UC-NRLF 
 
TREATISE 
 
 ON THE 
 
 THEORY OF THE CONSTRUCTION 
 
 OF 
 
 HELIC01DAL OBLIQUE 
 
 ARCHES. 
 
 BY 
 
 JOHN L. CULLEY, C- E. 
 
 REPRINTED FROM VAN NOSTRAND'S ENGINEERING MAGAZINE. 
 
 TvrMT? 
 
 OF TH 
 
 UNIVERSfi 
 
 
 NEW YOKK: 
 
 D. VAN NOSTRA^D, PUBLISHER, 
 23 MURRAY AND 27 WARREN STREETS. 
 
 1886. 
 
Copyright, 1886. 
 BY JOHN L. CULLEY. 
 
PREFACE. 
 
 I have attempted in this treatise to at- 
 tain two results : 
 
 First. To supply our American engi- 
 neering literature with a short, clear 
 treatment of the construction of Helicoi- 
 dal Oblique Arches ; and, 
 
 Second. To render simple all prob- 
 lems connected with their theory or con- 
 struction. I especially hope that I may 
 make the second plain to all who shall 
 read these pages. 
 
 Long since I have been satisfied that 
 much of the confusion and misunder- 
 standing arising from the attempts to 
 understand this subject, have arisen from 
 the fact that authors have failed, either 
 to state the fundamental principle, or to 
 keep it constantly before the student's 
 mind. Hence the general opinion has 
 arisen that helicoidal arches are of the 
 most intricate construction, and too often 
 
 115837 
 
IV 
 
 their consideration has been abandoned 
 with disgust. 
 
 The conception of a single principle 
 will clear away all this misunderstanding. 
 It is that of the process of the genera- 
 tion of helicoidal surfaces. It is a sim- 
 ple one, and, if constantly kept in mind, 
 will render all other problems equally 
 simple. No engineer's education is com- 
 plete without a thorough knowledge of 
 this subject. If the simple propositions 
 of Chapter I. are mastered, there will be 
 no trouble with the remainder of the 
 treatise. 
 
 Since the theory of Logarithmic 
 Arches is readily understood, after that 
 of helicoidal arches has once been thor- 
 oughly mastered, I have added a short 
 discussion of Logarithmic Oblique Arches. 
 
 JOHN L. CULLEY. 
 CLEVELAND, April, 1886. 
 
Treatise OB the Theory of the Construction 
 of Helicoidal Oblique Arches, 
 
 CHAPTER I. 
 
 ELEMENTARY PRINCIPLES. 
 
 1. A helix is a cylindrical curve that, 
 in passing over equal portions of the 
 circumference of the cylinder, travels 
 over the same number of equal portions 
 of the length of the cylinder. 
 
 2. Thus, in Fig. 1, A B C is the eleva- 
 tion of the semi-cylinder, A C D E, 
 whose diameter is A C, and length is 
 A E. Let A E also be the cylindrical 
 length of the semi-helix, A D, whose ele- 
 vation is A B C. 
 
 Divide A E and the circumference, 
 ABC, each into the same number of 
 parts of equal length, as shown in Fig. 1. 
 Through the points, 1, 2, 3, &c., of A E 
 draw lines parallel to A C, and through 
 the corresponding points, 1', 2,' 3', &c., 
 
of A B C, draw lines parallel to A E. 
 The intersections, 1", 2", tt", are points, 
 in the plan, of the semi-helix, A D. 
 
 Any number of points may thus be de- 
 termined, and the curve of the semi- 
 helix readily located, or drawn in the plan, 
 
Or, when the relation of the cylindrical 
 length to the semi-circumference is con- 
 stant, we can readily determine the equa- 
 tion of the curve, A D. 
 
 3. If the cylindrical surface, A C D E, 
 be straightened out into a plane surface, 
 its width will evidently be equal to the 
 semi-circumference, ABC, and its length 
 will be equal to A E, whilst the semi- 
 helix, by construction, would become a 
 straight line on this surface. Therefore, 
 in Fig. 1 draw A G equal to the semi-cir- 
 cumference, ABC, perpendicular to A E, 
 and complete the rectangle, A E F G. 
 This rectangle is called the development 
 of the semi-cylindrical surface, A C D E, 
 since it is its equivalent in plane surface. 
 If we regard A C D E the inner surface 
 or soffit of a right arch, A E F G is the 
 development of the intrados, and the 
 straight line, A F, is the development of 
 the intradosal semi-helix, A D. 
 
 4. From this we can determine the lo- 
 cation in the plan of a semi-helix I J nor- 
 mal to A D at L (Fig. 2). Its develop- 
 ment will evidently be also a straight 
 
9 
 
 line perpendicular to A F. Through L 
 draw L M parallel to A G, intersecting 
 A F at M. I M*H drawn through M per- 
 pendicular to A F, is the development of 
 the semi-helix normal to A D at L. Draw 
 H J parallel to A G, then J and I will be 
 the extremities of the normal semi-helix 
 in plan, whose cylindrical length is I K, 
 and elevation is ABC, whence we can 
 determine the J L I in the same manner 
 as A L D was determined, and as here 
 shown. 
 
 5. In Fig. 3 H I J is the elevation, and 
 H J K L the plan of the extrados or 
 outer surface of a right arch of the depth 
 I B, whose extradosal development is 
 H L M N, wherein H N equal to the 
 semi-circumference H IJ is drawn per- 
 pendicular to H L, and L M and M N are 
 respectively drawn parallel to H N and 
 toHL. 
 
 The extradosal and intradosal develop- 
 ments are both made from the axis X X for 
 the convenience of the comparison of these 
 surfaces and their lines. Here H P K is 
 the plan of the semi-extra dosal helix HIJ, 
 
10 
 
11 
 
 whose cylindrical length H L is equal to 
 A E of the semi-intradosal helix APD. 
 The curve H P K is determined from H L 
 and H IJ in the same manner as was 
 APDfrom AE and ABC. The straight 
 line H O M is the development of H P K. 
 
 By construction P and O, the respec- 
 tive points of intersection of the curves 
 H P K and A P D, and of the straight 
 lines H O M and A O F are both on the 
 line X X X'X', i. e. in plan and in devel- 
 opment. They are also on the same 
 line P O drawn at right angles to either 
 X X or to X' X'. Hence when either point 
 P or O is determined the other point is 
 also determined by the right angled line 
 OP. 
 
 6. The surface included between the 
 curves H P K and APD is called the 
 arch helicoidal surface, or arch helicoid, 
 and a careful examination of Fig. 3 will 
 elucidate to us the fundamental princi- 
 ple of helicoidal arch analyses and con- 
 struction. This fundamental principle is 
 that of the process of the generation of 
 this helicoidal warped surface. 
 
12 
 
 7. All lines 1 a IV, 2 b %b', &c., Fig. 
 3 drawn normal to the axis of the arch 
 and included between the intradosal and 
 extradosal helices are evidently elements 
 of, and the only straight lines lying en- 
 tirely within this warped surface. 1 a, 2 b> 
 &c., are the actual lengths of 1 a IV of 
 2 2'#', &c., since IV, 2'#', &c., produc- 
 ed are normal to the axis X X, or parallel 
 to the right section of the arch. They 
 are each equal to I B the depth of the 
 arch as they are the portions of the radii 
 x 1 a : x 2 b, &c., included between ABC 
 and H I J. 
 
 Again the radii XI a, X 2 b, &c., at their 
 points of contact in the in trades and in 
 extrados, are perpendicular to these sur- 
 faces, and are perpendicular to all lines 
 within these surfaces passing through 
 these points of contact, and are therefore 
 perpendicular to the helices passing 
 through these points. In other words, 
 any radius X 1 a or X 2 b drawn from the 
 axis XX to either intradosal or extradosal 
 helix, is at the point of contact with the 
 helix perpendicular to it. From this it 
 
13 
 
 follows that the elements la 1'a'; 2# 
 2'&', &c., are at the same time perpen- 
 dicular to both the in trad o sal and extra- 
 dosal helices A P D and H P K. 
 
 8. Whence, when the right sections of 
 the intrados and the extrados of an arch 
 are circular arcs, we derive the following 
 fundamental principle : 
 
 First. A helicoidal surface is generated 
 by a right line perpendicular to 
 and moving on the axis as one di- 
 rectrix and on a helix as the other 
 directrix, <ind the arch helicoid is 
 that part of this surface generated 
 by that portion of the generating 
 line, equal to the depth of the arch, 
 and included between the intrados 
 and the extrados. The said gen- 
 erator of the arch helicoid is at 
 all times perpendicular to the in- 
 tradosal and extradosal helices of 
 such helicoid. 
 
 And as a corollary, 
 
 The intradosal and extradosal helices 
 of an arch helicoid are parallel 
 
14 
 
 spiral curves whose perpendicular 
 distances apart equal the depth of 
 the arch. 
 
 9. These deductions are the essence of 
 all problems connected with this subject. 
 Nor should any one pursue it further 
 without a thorough mastery of the prin- 
 ciples here stated. A full understanding 
 of the process of the generation of heli- 
 coidal surfaces is of the first importance, 
 and such a conception will render all fur- 
 ther consideration of the elements of 
 these warped surfaces clear and simple. 
 
 CHAPTER II. 
 
 HELICOIDAL CORVES AND TEMPLETS TWIST 
 RULES. 
 
 10. In the plan (Fig. 4) of the intra- 
 dosal helix A P D, make P S' equal to P 
 R', and join R' and S' by a straight line, 
 R' S'. R' S', by construction, will pass 
 through P. R S will be the elevation of 
 the chord joining the extremities of the 
 helix R' S'. If, then, we lay off R" S'' 
 
15 
 
16 
 
 equal to and parallel toR' S', we can con- 
 struct on it the curve of the intersection 
 of a right vertical plane, R' P S', with 
 the intrados, A C D E. It will be evi- 
 dent, by reference to the elevation, A B 
 C, that the middle ordinate X 4 of R" 
 S" at X, will be equal to X B in eleva- 
 tion. Again, if at U and T in the ele- 
 vation we draw perpendiculars to R S, 
 these perpendiculars will respectively 
 equal the ordinates V 3 and W 5 of R" S" 
 at V and W. In this manner any num- 
 ber of points, 3, 4, 5. &c., may be obtained 
 and the curve, S", 3, 4, 5, &c., R'' con- 
 structed, which will be the curve of the 
 intersection of the vertical plane, R' S', 
 with the intrados, A C D E. 
 
 11. The corresponding extradosal ellip- 
 tic curve, 678, may be similarly con- 
 structed, or both it and the intradosal 
 curves may be determined for the equa- 
 tion of their ellipses. 
 
 12. Let R" R' S' S" (Fig. 5) be the 
 plan and S'', 3 4 5 R" the elevation of a 
 piece of timber of any thickness, R"' R'. 
 On R" R' S' S" produce R' P S' of the 
 
17 
 
 plan of the helix in Fig. 4 and at any con- 
 venient distance from, and parallel to, 
 R' P S' draw the dotted line here shown. 
 
 The space between R' P S' and the dot- 
 ted parallel line will be the plan, and 
 S" 3 4 5 R" the elevation of a templet of 
 the intradosal helix, or the soffit coursing 
 joint, RBS-R'PS'. 
 
 Care should be observed that R' R" 
 and S' S" in the plan of this templet are 
 made parallel with the axis or spring 
 lines of the arch, and that the spiral R' P 
 S' and the dotted curve here shown are 
 made parallel to one. another on lines 
 parallel with R' R" and S' S". 
 
 In like manner the templet of the ex- 
 
18 
 
 tradosal helix may be constructed. When 
 any considerable portion of a helix is to 
 be treated at one time this is the true 
 way to construct the coursing joint temp- 
 lets. 
 
 TWIST KULES. 
 
 13. Let Fig. 6 be an enlarged drawing 
 of the plan and development of the heli- 
 ces, A P D and H P K, between the point, 
 P O, and the element, 1' a' (Fig. 3). It 
 will be observed that the point, P, is 
 the plan of a vertical element of the arch 
 helicoid, or the element at P is perpen- 
 dicular to the horizontal plane. The 
 element P and the point, 1', of the intra- 
 dosal helix, locate the vertical plane pass- 
 ing through P and 1', whose trace on the 
 horizontal plane is P 1'3'. Then the line, 
 a'3', drawn from a' in the extradosal he- 
 lix perpendicular to the vertical plane, 
 P' 1' 3, is the measure of the warp of the 
 arch helicoid at 1' a' 1 a' from the plane, 
 P 1' 3', for the given depth, I B, of the 
 arch. 
 
 14. 1' a is, by construction, equal to 
 I B and a' 3' being perpendicular to the 
 
19 
 
20 
 
 vertical plane, is perpendicular to any 
 line within the vertical plane passing 
 through its foot, 3'. Consequently a 3' is 
 perpendicular -to the line whose plan is 
 V 3'. 
 
 Therefore on the straight line, 4 5 (Fig. 
 7) erect the perpendicular, 5 6, equal to a'3', 
 
 6 ir 
 
 Fig, 7 
 
 14 
 
 15 
 
 and with 6 as a center, intersect 4 5 at 4 
 with an arch whose radius, 4 6, equals 
 I B, the depth of the arch. 4 5 6 is the 
 actual size of the wedge, V 3' a', in plan 
 
21 
 
 that is perpendicular to the vertical plane, 
 P V 3', and that supports the plane above 
 the helicoid, P 1' a' at V a . It should 
 be noticed that 4 5 is the actual length 
 of the line, 1' 3', in the plan or in the 
 trace of the vertical plane. 
 
 15. To determine the warp of any ele- 
 ment, 2' b f , from b' and from 2' draw the 
 perpendiculars, b' 1 and 2' 8 to P 1' 3'. 
 Then on any line, 9 10, erect a perpendicu- 
 lar, 10 11, equal to b' 7, and lay of 10 13, 
 equal to 2' 8, and through 13 draw 12 13 
 parallel to 9 10. Then, with 11 as a cen- 
 ter, intersect 12 13 at 12 with an arc of 
 the radius, 11 12, equal to I B, and draw 
 9 12 parallel to 10 11. 9 10 11 12 will 
 be the actual size of the truncated wedge 
 shown in plan at 2' b' 1 8, that is perpen- 
 dicular to the vertical plane through P 
 and 1', and that supports this plane 
 above the helicoid at 2' b 2 b. 
 
 In same manner the warp of any ele- 
 ment of the helicoid from the vertical 
 plane, P V 3', may be obtained. 
 
 16. The figures 456; 9 10 11 12, and 
 a straight line, 14 15, equal to I B, are 
 
22 
 
 templets of the helicoid respectively at 
 1' a' -I a : V V -2 6, and at P-O, and 
 in applying them care should be exercised 
 that the lines 4 6 and 11 12 are applied to 
 the helicoid with the points 4 and 12 ex- 
 actly at 1' 1 and at 2' 2, and with the 
 points 6 and 1 1 exactly at a' -a and at #'&. 
 The lines, 4 5 and 9 10 are, of course, to be 
 brought exactly into the plane, P' 1' 3', 
 while the straight line, 14 15, will be ap- 
 plied at the intersection of the plane 
 P I' 3', and the helicoid at P O. In 
 practice, however, a straight line, 14 15, 
 cannot be used. Therefore, draw 16 17 
 parallel to 1415, 1417 distinct therefrom, 
 and increase the depths of 4 5 6 and of 
 9 10 11 12 each a distance equal to 14 17 
 to the dotted lines shown. 
 
 Then 16 17 and the dotted lines, 4 5 
 and 9 10, will be in a plane parallel to the 
 plane P 1' 3', and these parallel and twist 
 rules should, when applied, be always 
 perpendicular to this parallel plane that 
 they may ever be perpendicular to the 
 first plane, P 1' 3'. It is very important 
 that these rules in application should 
 
23 
 
 always be maintained normal to the sight 
 plane. Otherwise it would be impossible 
 to reduce true and correct warped sur- 
 faces. 
 
 The parallel and twist rules are all 
 known as twist rules, as they are the 
 templets by which the warp of the cours- 
 ing beds are worked. 
 
 17. Fig. 8 is a perspective view show- 
 ing the manner of applying the twist and 
 
 parallel rules of Fig. 7. 15 11 6, is the 
 extradosal helix whose exact length and 
 that of its parts, 15 11 and 11 6, are 
 taken from the development (Fig. 6), 
 and are respectively equal to O a, O b 
 and b a. 14 12 4 is the intradosal 
 helix equal to 0, 1, and its parts 14, 12 
 
24 
 
 and 12, 4 are equal to 02 and 21 (Fig. 
 6). The parallel and twist rules are 
 of the same size and characteristics 
 as in Fig. 7. Their upper edges 16 17, 
 9 10, and 4 5 lie in and coincide with 
 the plane 4 5 16 17, to which these rules 
 are perpendicular. The corners of the 
 rules 6 11 and 15 are in the extradosal 
 helix 6 11 15, while the corners 4 12 and 
 14 are in the intradosal. It will also 
 be noticed that since the lower edges of 
 these rules coincide with elements of the 
 helicoid, they are and should be normal 
 to the two helices of the helicoid. 
 
 18. We have here used in Fig. 8 three 
 rules simply for the convenience of illus- 
 tration. In practice, many may be used. 
 Sometimes a parallel rule and a single 
 twist rule will be all that are needed, and 
 in fact this will generally be the case- 
 The number of twist rules to be used de- 
 pends upon the length and warp of the 
 Toussoir treated. However, enough of 
 these should always be used to exactly 
 determine the warped helicoids. 
 
 19. This method of determining the 
 
26 
 
 warped helicoids will overcome Professor 
 Hyde's objection to the ordinary method 
 of determining them, referred to in his 
 book, " Skew Arches," pp. 11 and 12. 
 
 20. John Watson Buck in his treatise 
 on this subject, " Essay on Oblique 
 Bridges," pp. 13 and 14, presumes to 
 determine the warp of the helicoids in 
 the following manner : Let P 1' a' and 
 Ola (Fig. 9) be the plan and develop- 
 ment of a helicoid. From 1 in the develop- 
 ment, draw 1 3 perpendicular to O a. It 
 is there stated that 1 3 is the warp of 
 the element 1'a' from the plane P 1' 3'. 
 It has already been proven equal to a' 3'. 
 Therefore a' 3' would have to be equal to 
 13'. But by construction they are not 
 equal. Buck's rule is not strictly true. 
 It seems to be an empirical rule not 
 demonstrable. But inasmuch as the 
 length of the voussoirs are near]y always 
 small in comparison to the whole length 
 of the helix, this rule, though not theo- 
 retically true, is practically correct ; for 
 at the points P and O the curved lines 
 1' 2' P and a b r P, and the straight lines 
 

 27 
 
 120 and a b make the same included 
 angles. 
 
 But if warped helicoids of any magni- 
 tude are to be worked at one time, it 
 would not be safe to use Buck's rule. 
 The rule we have given is true for any 
 length of the helicoid, and as easily ob- 
 tained and applied as Buck's, and there is 
 no reason why it should not be always 
 used. , (See art 62.) 
 
 21. Produce the lines I' a f and 1 a (Fig, 
 10) to c and d in XX and X'X' 
 
 Let the /3 be angle of the intrados, 
 1 O d'\ j3 f be the angle of the extrados 
 a O d r (Fig. 10) ; # equal the angle V P c 
 Bangle 3 VI' by construction; r tlie ra- 
 dius of the intrados and / the radius of 
 
 1 d a d 
 
 the extrados, = -. 
 nr 1 
 
 Then 
 
 ?c=Od=O 1 cosin /3=Oa cosin ft' (1) 
 
 ld=Odifm. ft . . (2) 
 
 and ad=Odi&u. ft' . (3) 
 
28 
 
29 
 
 . (4) 
 . (5) 
 
 Whence we obtain 
 
 cV 
 Vaca'c\, and tang. #=-=j- . (6) 
 
 JL C 
 
 .-. 3'a'=I'a sin. < . (7) 
 In like manner any warp distances b f 1 
 and 2' 8 may be obtained. 
 
 CHAPTER III. 
 
 OBLIQUE ARCHES HELICOIDAL AND OTHER 
 
 METHODS. 
 
 22. It will often happen in public im- 
 provements and in architectural con- 
 structions that arches are so located that 
 their parallel ends or faces cannot be 
 placed normal to axes of the arches. 
 Such arches are called Oblique or Skew 
 Arches, and the acute angle the axis 
 makes with the plan of either face of the 
 arch is the angle of the obliquity of the 
 arch. 
 
30 
 
 23. The stability of an arch where the 
 coarse stones or voussoirs are constructed 
 in the ordinary manner, with their cours- 
 ing joints parallel with the axis, is weak- 
 ened the moment the plane of the arch 
 faces ceases to be normal to the axis. 
 This weakness increases as the angle of 
 obliquity decreases. 
 
 24 Thus let Fig. 11 be the plan and 
 elevation of a semi-circular oblique arch, 
 whose coursing joints are parallel to its 
 axis X X. Through H and K draw the 
 perpendiculars H 1 and K 2. The thrust 
 of the arch due to its weight and load 
 would naturally be carried to the abut- 
 ments in lines parallel to H 1 and K 2. 
 But the triangles H J 1 and L K 2 are 
 each supported by a single abutment. 
 They therefore depend upon the bond 
 and strength of the voussoirs within 
 them and in the body of the arch imme- 
 diately beyond them, to direct a part of 
 their thrust and stress to the opposite 
 abutments. Moreover, the thrust and 
 stress thus conveyed must necessarily be 
 in lines or planes other than those par- 
 
31 
 
 allel to H 1 and K 2, which would cause 
 an outward movement in these triangles? 
 which would be greatest at or near the 
 
 J C 
 
 A H 
 
 ig. 11 
 
 point J and L. When the angle of ob- 
 liquity H J 1 or K L 2 is very acute, 
 neither the bond or the strength of the 
 voussoirs will resist the tendency to rup- 
 
32 
 
 ture, nor would the friction of the bed- 
 ding surfaces produced by the weight 
 and load of the voussoirs prevent the 
 outward movement at J or L, and the 
 arch will fail from sheer weakness. 
 
 25. It is true that small oblique arches, 
 with their coursing joints parallel to 
 their axles, are often with safety built. 
 Their stablity is due to the fact that as 
 ordinarily constructed the depth of their 
 voussoirs is in excess of what economy 
 would in larger arches dictate. In other 
 words, the voussoirs maintain their posi- 
 tion by virtue of their large bedding 
 frictional surfaces. If, however, economy 
 or other consideration should require 
 their depth of ring to be minimum, or if 
 the parts H J 1 and L K 2 were loaded 
 anything like the body of the arch is 
 capable of sustaining, the smaller like 
 the larger arches would give away. 
 
 26. It is evident that to maintain the 
 stability of an oblique arch it is desirable 
 that its stress and thrust should be con- 
 veyed to the abutments in planes paral- 
 lel to the oblique faces of the arch, or as 
 
33 
 
 near so as is possible, and also that 
 this object cannot be attained when the 
 coursing joints are parallel to the axis 
 of the arch. The coursing beds to fully 
 realize this condition should be normal 
 to the oblique faces of the arch wherever 
 they come in contact with them. 
 
 27. There are three methods in vogue 
 that closely approximate this require- 
 ment. They are known as the Logarith- 
 mic, the Corne de Vache, or Cow's Jlorn, 
 and as the Helicoidal Methods. 
 
 Without entering into the comparative 
 merits of these three methods, it will be 
 sufficient for the present purposes to 
 state that the main advantages of the 
 Helicoidal Method are : 
 
 The successive coursing beds and joints 
 are parallel to one another throughout 
 their entire lengths, rendering all vous- 
 soirs of the same Warp in their coursing 
 beds, their soffit and in their extrados ; 
 also in their bedding surfaces, except in 
 the arch face. Therefore, all voussoirs 
 of the same length are exactly alike, 
 rendering only one set of templets neces- 
 
34 
 
 sary in their execution a condition not 
 realized in tbe other two methods, for 
 in both of them each stone is always dif- 
 ferent from the next one to it in the same 
 course. This consideration materially 
 reduces the number and expense of meas- 
 urements, drawings and templets. For, 
 in the two other methods each stone has 
 its separate and distinct templets, meas- 
 urements and drawings, and are the ob- 
 jects of much care and anxiety. Thus it 
 is a helicoidal voussoir will fit into any 
 portion of its course, not requiring to be 
 set at a particular place in its course as 
 is required by the other methods. Again, 
 the ring stones at equal distances from 
 the key or center of the arch face, being 
 of the same dimensions this method pre- 
 serves the pleasing architectural effect of 
 an arch, without the distressing effect 
 produced by the courses increasing or 
 decreasing in size from one side of the 
 arch to the other as is done by the other 
 methods. 
 
 28. Whatever may be said as to the 
 stability of oblique arches by the other 
 
35 
 
 methods, it is true helicoidal arches with 
 very acute angles of obliquity may with 
 safety be constructed. This acuteness 
 can be considerably less than 30. John 
 Watson Buck, M. Inst. C.E., places its 
 minimum value at 25. But it is proba- 
 ble that if the skew backs are maintained 
 in their place that this value may be even 
 less. 
 
 An oblique arch will, of course, give 
 away from defective construction, as for 
 instance an insecure abutment, a weak 
 foundation, or a lack of proper bond, 
 just as a right arch would under like cir- 
 cumstances. Yet it is quite the custom 
 to condemn the theory of correct con- 
 struction of oblique arches, when the 
 weakness in the arch was in no way due 
 to the arch proper, either in theory or 
 construction. To be secure an arch in 
 all its parts it must be well and thoroughly 
 built. I have known oblique arches that 
 stood better than could have been ex- 
 pected of right arches under like condi- 
 tions, simply because the oblique arches 
 were unusually well built. 
 
36 
 
 29. Professor Hyde, in his treatise 
 heretofore referred to, page 101, states 
 that an advantage of this method is 
 owing to the fact that the coursing joints 
 are parallel, that the courses or whole 
 arch, with the exception of the ring 
 stones, may be constructed of bricks, 
 which is very true. Great caution, how- 
 ever, should be exercised in such con- 
 struction. The crushing strength of or- 
 dinary bricks to that of ordinary building 
 stone is as 1 to 7, and arches never should 
 be built of bricks, unless they are of a 
 first-class, hard burnt quality. This is 
 especialiy true in public works exposed 
 to the action of the elements. It would 
 be better and safer to recommend that 
 the arches between the ring stones be 
 made of thin coursed rubble of parallel 
 beds, which material is equally adapted 
 to helicoidal construction. 
 
 30. Undoubtedly many oblique arches 
 have been thus constructed of brick^ 
 either from a want of knowledge of the 
 principles of oblique arch construction, 
 or from erroneous ideas of the excessive 
 
37 
 
 cost of warped stone voussoirs over that 
 of straight voussoirs for the same arch. 
 This excessive cost is but a small per 
 cent, over right arch construction, as will 
 be proved further on. In oblique arch 
 construction brick work is undoubtedly 
 cheaper than dressed stone work, but is 
 not much cheaper than coursed rubble 
 work, while the latter is far more stable, 
 but neither will compare with dressed 
 stone for strength. 
 
 Brick or coursed rubble work can be 
 used to great advantage in the arch be- 
 tween the dressed end ring stones, when 
 the axis and spring lines of the arch are 
 parallel curves. Much less care is nec- 
 essary in arranging the courses when 
 laid up of brick or of coursed rubble. 
 
 HELICOIDAL METHOD. 
 
 31. The helicoidal method of construc- 
 tion of oblique arches is one in which 
 the coursing beds of the voussoirs are 
 radial helicoids, whose intradosal and ex- 
 tradosal helices are parallel curves. 
 
 32. A helicoid, as stated in article 8, 
 
38 
 
 J C 
 
 A H 
 
 W 
 
 is generated by a right line perpendicu- 
 lar to, and moving on, the axis of the 
 arch as one directrix, and on a helix as 
 
39 
 
 the other directrix, and the arch helicoid 
 is that part of this surface generated by 
 that portion of the generating line that 
 is equal to the depth of the arch, and is 
 included between the intrados and the 
 extrados. 
 
 33. J W U H in Fig. 12 is the plan of 
 an oblique arch ABC, and H K J equal to 
 K W U is the angle of obliquity, and 
 the helicoid H P K A P D represents a 
 coursing bed of one course of voussoirs. 
 
 34. The extrados is removed in the plan 
 of Fig. 13 in order to show the intra- 
 dosal helices of the successive courses 
 and the traces 1 2, 3 4, &c., of the heli- 
 coids or coursing beds upon the imposts 
 A H and C J of the abutments, while in 
 Fig. 14 these traces 1 2, 3 4, &c., and the 
 corresponding extradosal helices, are 
 shown. The traces 1 2, 3 4, &c., are al- 
 ways normal to the axis of the arch, be 
 the arch segmental or full semi-circular, 
 since the generator of the helicoids is 
 radial and therefore normal to the axis of 
 the arch. These traces are of course 
 
40 
 
 A H 
 
 inig.13 
 
 elements of their several helicoids as well 
 as of the imposts. 
 
41 
 
 Were it not for the confusion of 
 curved lines, both the intradosal and ex- 
 tradosal helices and the traces 1 2, 3 4, 
 &c., might be given in the same plan, 
 and each helicoid or coursing bed would 
 be shown as in Fig. 12. Great con- 
 fusion would result from such an at- 
 tempt. Fig. 12, therefore, will suffice 
 for a comparison of the helices of the 
 several helicoids. 
 
 CHAPTER IV. 
 
 THE FOREGOING PRINCIPLE APPLIED TO HELI- 
 COIDAL OBLIQUE ARCHES AND DE- 
 DUCTION OF FORMULA. 
 
 35. In the application of the foregoing 
 principles to helicoidal oblique arches a 
 constant relation is established for any 
 given case between the right diameter of 
 the arch and the length of either the in- 
 tradosal or extradosal helix of the arch 
 helicoid. The position of the intradosal 
 helix is then determined by drawing it 
 perpendicular to a straight line, joining 
 
42 
 
 impost extremities of the arch face end of 
 the developed intrados. Thus in Fig. 16 
 
44 
 
 the developed intradosal helix S. F. is 
 made perpendicular to the straight line 
 S X' G. As there is a constant relation 
 between the length of the intradosal and 
 extradosal helices, this construction would 
 also establish a constant relation of the 
 diameter of the arch to the length of the 
 extradosal helix. 
 
 Again, it should be observed that since 
 the skew face of the arch in plan is a 
 straight line, the development of either 
 the intradosal or extradosal arch face 
 ellipse will be a curved line, and that the 
 straight line in the development joining 
 the impost extremities of the ellipse will 
 divide this curve line into equal curves. 
 
 36. Thus, let A B C - H I J (Fig. 16) 
 be right section or elevation of an oblique 
 arch, and S C V U the plan of its intrados. 
 Draw A G equal to the semi-circumfer- 
 ence ABC perpendicular to A E and 
 divide A G and ABC into the same con- 
 venient number of parts of equal length 
 and draw the straight line G S. It is 
 evident that the middle point of the 
 ellipse S C in the plan will, in the devel- 
 
45 
 
 opment, be the middle point of G S. 
 Then, through the points of division in 
 ABC and A G draw lines parallel with 
 AE, and through the points of intersection 
 with S C draw lines parallel with A G, the 
 points of intersection of these lines the 
 parallel with A G, with the corresponding 
 lines parallel with A E, drawn through 
 the points of division in A G, will be 
 points in the curved line S X' G. 
 
 Through S draw !^ F perpendicular to 
 the straight line S G. S F will be the 
 development of the helix S D, whose 
 cylindrical length is S E, and from S F 
 and S E the curve S D may be obtained 
 according to article 2. The angle F S E 
 equal to S G A by construction, is the an- 
 gle of the intrados and is designated by /?. 
 
 37. In Fig. 16 the intrados and ex- 
 trados of the arch are shown in plan, 
 elevation and in development. The ex- 
 tradosal helix K K is determined from 
 the intradosal helix S D by drawing 
 through S and F, S E and F M perpen- 
 dicular to E L and to M N and joining 
 E and M by E M. Then E M will be the 
 
46 
 
 development of the helix E P K, whose 
 cylindrical length E L equals S E of the 
 intradosal helix S P D. From E L and 
 E M the curve E P K in the plan can be 
 obtained in the same manner that S P D 
 was. Also the curved end E X' N is pro- 
 jected in the development similarly for 
 H N and H IJ as S X' G was from A G 
 and ABC. The angle M E L is the 
 angle of the extrados and is designated 
 
 38. Let r be the right radius of the 
 intrados ; r' the radius of the extrados, 
 b the depth of the arch, and the com- 
 plement of the angle of obliquity. Then 
 n r=A. B C=A G, and it r r^H I J=H N. 
 Then the relation of the diameter or ra- 
 dius to the helices will be obtained from 
 the following formulae : 
 
 AS=2r tang. d^Ttr tang, ft (8) 
 .-. tang, p . (9) 
 
 This eq. is true and ft bears this rela- 
 tion to 6 so long as S F is normal to the 
 straight -line S G. Sometimes it will be 
 
47 
 
 necessary to alter this relation of S F and 
 S G, so that S F will pass through 'i given 
 point in U Q. This alteration is always 
 slight, and eq. ^9) can readily be correct- 
 ed for it. 
 
 CS cos. 6=2r. or 08 = ^ (10) 
 
 cos. u 
 
 7t T 
 
 GS cos. ftrrr, or GS = rr(ll) 
 
 cos. p 
 
 Again (E F = 7t r) = S E tang. /?, 
 7t r n r' n* r 
 
 "tang, ft tang./?' 2 tang. 9 
 
 s5 (13 > 
 nr' 
 
 (14) 
 
 39. In Fig. 16, let X X be the axis of y 
 and P O that of x, with the origin at P, 
 and the equation of the curve S P D with 
 reference to these axes, will be obtained in 
 the following manner : 
 
 Let 2 be any point in S P D, whose 
 ordinates are x and y, the angle 3 X B be 
 a, and n be the quotient of the length 
 
48 
 
 of the arc B3, divided by the length 
 of the semi-circumference A B^C, then 
 a=n 180, x=r sin. a, 
 
 n*r x 2ytan. H 
 
 =n TT-r - a .'. r=- - =s= -^ - (15) 
 2 tang. 6 sin. a twrr 
 
 (15) is the equation of the curve S P D, 
 so that if either x or y is given, n and y 
 or x can be obtained. It is also the 
 equation of the plan of the normal helix 
 to S P D at P, when y equals the nib. 
 part of the normal helix's cylindrical 
 length, or n K I in Fig. 2. 
 
 40. To determine the equation of K P K 
 let x and y be the ordkiates to any point 
 4 in it, referred to same axes as^in article 
 39. Then, as before, 
 
 but x=r sin. a=(r + b) sin. a 
 
 n 7t sm. a 
 
 41. To determine the equation of the 
 
 end curve S X' G of the development 
 
 with reference axis 5 X' 6 of a*, draw n 
 
 through X' parallel to A G and axis of y 
 
49 
 
 X' X', let x and y be the ordinates of any 
 point y in S X' G. Then by reference to 
 Fig. 16 it will appear 
 
 x 
 
 x=n re r or r= 
 
 n : 
 
 d tang. 6 or r=^ 
 
 y 
 
 sin. a tang. 6 
 (17) 
 
 ' n n sin. a tang, d 
 
 42. And for the corresponding extra- 
 dosal end curve K X' N we have as in 41 
 
 , , x 
 x=n n r or r = 
 n n 
 
 y=r sin a tang, dorr' 
 
 sin. a tang. 
 (18) 
 
 ' n n sin. a tang. 6 
 
 43. In practice it will be impossible to 
 make full-sized drawings of the curves 
 S P D, E P K, S X' G and R X' N for 
 their entire lengths. The four equations 
 above will be useful in exactly determin- 
 ing any portion of either one of them, and 
 be of great assistance in laying out the 
 work for construction. 
 
50 
 
 44. In a properly constructed arch the 
 resultant of all faces acting upon it should 
 be confined within the middle third of 
 the depth I B of the arch. 
 
 When S F in the development is per- 
 pendicular S G, R M cannot be perpen- 
 dicular to the line joining the impost 
 extremities of arch face ends of the de- 
 veloped intrados. If then for any reason 
 it is desirable to alter the direction of 
 S F it should be so altered whenever 
 practicable, that the dotted line midway 
 between S F and K M (Fig. 16) should ap- 
 proach to or become normal to the straight 
 line, joining the extremities of the develop- 
 ment of the middle line of the face of the 
 arch here shown by dotted line in the ele- 
 vation midway between ABC and H I J. 
 
 45. In the intradosal development the 
 coursing joints are laid out in the follow- 
 ing manner. Let S U Q G (Fig. 17) be 
 the intradosal development of an arch, 
 and let E T and N W be the outer edges 
 of the extradosal development. Through 
 S draw S F perpendicular to the straight 
 line S G. Divide the straight lines S G 
 
51 
 
 and U Q each into an odd number of 
 parts of equal length in order to show a 
 key in the arch face. If S F should not 
 
52 
 
 pass through one of the points of division 
 in U Q the length of the arch should be 
 increased or decreased, or altered the di- 
 rection of S F as indicated in the last ar- 
 ticle, so that it will pass exactly through 
 one of the points of division in U Q. 
 Then draw lines through the points of 
 division in S G and U Q parallel to S F. 
 The portions of -these parallel lines be- 
 tween the ends K G and U Q will be the 
 intradosal coursing joints of the succes- 
 sive courses in the development. The 
 soffit face between them may then be 
 divided into convenient lengths, as shown 
 by drawing their heading joints at right 
 angles to them. 
 
 46. Let 1 T W N, Fig. 18, be the cor- 
 responding extradosal development to 
 Fig. 17, showing the springing edges S U 
 and G Q of the extrados. Through S draw 
 S E perpendicular to S U, and lay off the 
 spaces E 2, 2 3, 3 4, &c., Fig. 17, on E T. 
 Then through E draw EM parallel to 
 E M, Fig. 16, or as altered by Fig. 17, 
 and through 234, &c., and the corre- 
 sponding points in N W, draw lines par- 
 
53 
 
 allel to K M. Then divide the remaining 
 spaces of straight lines 1 N and and T W 
 into equal parts equal to the distance 
 
54 
 
 6 7 on T W, and through the points of 
 division draw lines parallel to R M. These 
 parallel lines will be the extradosal cours- 
 ing joints in the development. 
 
 CHAPTEK V. 
 
 METHOD OF WORKING THE VOUSSOIRS AND 
 SKEW-BACK STONES. 
 
 47. It should be observed of the sev- 
 eral faces of a voussoir that, while its 
 soffit and extrados are warped curved 
 surfaces, the bedding surfaces are ra- 
 dial warped surfaces. On this account 
 and because they are generally consider- 
 ably wider than the soffit faces, the bed- 
 ding courses are the most presentable 
 for the first surfaces to be worked. 
 
 48. The curve of the templets of the 
 intradosal and extradosal coursing joints 
 may be obtained exactly by the method 
 described in Article 12. In practice, 
 however, the elliptical curve, S"3.4.5R" 
 (Fig. 4.), will be sufficiently exact; for 
 when the length of the voussoir is small 
 
55 
 
 in comparison with the length of the 
 whole semi helix, the curve will vary 
 but slightly from this elliptical curve. 
 Very sharp oblique arches of small diam- 
 eter have been successfully built, where 
 this curve has been regarded as a circular 
 arct. The method of determining such 
 circular arcts is described farther on, in 
 Articles 52 and 53. The elliptical curve 
 is, of course, nearer the true spiral curve 
 than the circular arcs are. But neither 
 of these two approximate curves should 
 be employed unless it is found, after 
 careful examination not to materially de- 
 part from the true curve. 
 
 49. Let A B C D E (Fig. 19) be the ele- 
 vation, and E D the plan of a templet of a 
 soffit coursing joint, so that the curve 
 ABC between the center lines of the 
 iron strap, 1 and 3, will be the exact 
 length of a voussoir soffit coursing joint. 
 We will suppose the rules shown in Fig. 
 7 are the proper ones for the joint ABC 
 at points on it equidistant apart, and that 
 the voussoir bed warps from A H towards 
 C J, the point J being the point in the bed 
 
56 
 
 G 
 H 
 
 farthest from a plane passing through 
 the points A C and H. Evidently the par- 
 allel rule should be applied at A and the 
 twist rules at B and C, so that their sides 
 will be normal to, and their upper edges 
 shall coincide with the sight plane 4 5. 
 The iron straps 1, 2 and 3 are fastened 
 
57 
 
 to the templet ABODE so that their 
 center lines here shown will be normal to 
 the curve ABC, but their sides are nor- 
 mal to the plane 4 5. Their tops, like 
 their rules, coincide with this plane 4 5. 
 Let each of the three rules above referred 
 to, at their intradosal points A, B and C, 
 be extended over the templet A B C D E in 
 an arm that will exactly fit into its respec- 
 
 Fi s . 20 
 
 tive strap 1, 2 or 3 (Fig. 20). Thumb screws 
 in the top of iron straps will prevent the 
 
58 
 
 templets moving when they are once ad- 
 justed to these straps. The cross section 
 of the parallel and twist rules should be 
 of the form shown at K L in order that 
 its lower edge may occupy the least 
 space when applied to the voussoir bed- 
 ding surface. 
 
 50. Therefore, in working the coursing 
 bed of a voussoir, apply the soffit joint 
 templet (Fig. 19) to the stone from which 
 the voussoir is to be cut and on the sur- 
 face of the stone under which the coursing 
 bed is to be worked. It should be applied 
 to the stone far enough from the edge 
 of the stone on this surface to allow 
 for the working of the soffit surface 
 of the voussoir in the stone afterwards. 
 When the surface of the stone has been 
 dressed off to receive the templet let the 
 curve A B C be marked upon it. Cut a 
 narrow channel across the stone at A H 
 so that its bottom surface shall exactly 
 receive the lower edge of the parallel 
 rule when properly adjusted in its strap 
 1, Fig. 20, and the soffit joint templet 
 coincides with its curve already marked 
 
59 
 
 on the dressed surface of the stone. 
 Then cut a narrow channel across the 
 stone at B I, so that its bottom surface 
 will exactly receive the lower edge of its 
 twist rule, when this and the parallel 
 rule, properly adjusted to the soffit joint 
 templet, are applied at B I and A H re- 
 spectively. In the same manner the 
 third rule is applied at C J and the bot- 
 tom of its narrow channel reduced to re- 
 ceive the lower edge of this twist rule. 
 Care should be exercised in applying the 
 twist rules that their upper edges are al- 
 ways in the same plane and that the upper 
 edge of the parallel rule is in this plane, 
 all the rules, moreover, should be normal 
 to this plane Any number of twist rules 
 maybe thus applied. Ordinarily, a single 
 pair of one parallel and one twist rule 
 will be enough. The balance of the bed- 
 ding surface may be reduced to the bot- 
 tom surfaces of the grooves thus cut by 
 applying a straight edge to them on lines 
 parallel to A B and to B C. 
 
 Having thus determined the bedding 
 surface the extradosal curve H IJ may 
 
60 
 
 be drawn upon it parallel to and distant 
 the depth of the arch I B, Fig. 19, from 
 the intradosal curve ABC. 
 
 51. The ordinary method of making 
 these bedding surfaces of the voussoirs 
 is : 1st, lay off and sink the soffit course 
 joint ABC. Then the extradosal curve 
 H IJ is worked on the rough surface of 
 the stone, and the parallel A H and the 
 twist B I rule are applied normal to the 
 curve ABC, until the upper edges are 
 in the same plane. Then the twist rule 
 C J is similarly applied. Thus used the 
 rules have no arm over the soffit joint 
 templet in the iron straps, as in Figs. 19, 
 20 and 21, to retain them in correct 
 positions. Obviously this method of 
 reducing these surfaces is attended with 
 uncertain results, and that the method 
 described in the last article is far supe- 
 rior to it. The method there described 
 is true and exact, giving to all these sur- 
 faces the same warp or twist, a condition 
 that should always be maintained, and 
 for this reason the method of Article 50 
 should always be used. 
 
61 
 
 52. Care should be exercised that the 
 warp is worked in the right direction. 
 Nor should we be deluded by the sup- 
 
62 
 
 position that if the voussoirs are warped 
 the wrong way, they can be used in the 
 other end of the arch from that for which 
 they were intended. All voussoirs in a 
 given oblique arch have the same warp, 
 and therefore those that are warped the 
 wrong way cannot be used. This fact 
 should be noted, all oblique arches here 
 given have been left handed oblique 
 arches, and the parallel and twist rules 
 have been applied accordingly. When 
 the oblique arch is right handed the 
 order is reversed in their application. 
 
 53. We will now proceed to the meth- 
 od of working the warped soffit surfaces 
 of the voussoirs. In Fig. 22, let 5 6 7 8 
 be the plan, and 5' & T 8' the develop- 
 ment of the intrados of a voussoir, so 
 that the axis X X X' X' passes through 
 the middle points of the joints 5 6 5'6', 
 and 67 6T, and let the curves 6B7 
 and 5 E 6 immediately above and below 
 5678 each be circular arcs of the right 
 section of the intrados. In the develop- 
 ment the heading joint 6'7' is perpen- 
 dicular to coursing joint 5' 6' at 6', there- 
 
63 
 
 
 
 T" 
 
 CO/ j 
 
 
 
 
 
 a ^-^ 
 
 
 
 
 fl 
 
 
 
 
 
 01 
 
 1 
 
 CO 
 
 /\ 
 
 
 
 
 5 
 
 ^x 
 
 /^ \ 
 
 \ / 
 
 / 
 
 4 
 
 J l\ 
 
 ^^ 
 
 \ / 
 
 OS NJ 
 
 / 
 
 
 1 
 
 
 "x^ 5 I/ 
 
 / 
 
 
 
 X V> 
 
 
 8 
 
 
 I 
 
 : \ 
 
 v^ x \ 
 
 \ 
 
 
 c 
 
 \ c 
 
 
 
 \ 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 c 
 
 
64 
 
 fore these two joints are normal to one 
 another at the point of their intersection 
 66', and also to the element which is 
 common to the course helicoid and to the 
 heading helicoid at 66'. 
 
 The actual lineal curves of the cours- 
 ing joints 56 8 6' and of the heading 
 joints 6 7 6' 7', or their elliptical approxi- 
 mates can be determined by the methods 
 of Article 10. When regarded as arcs of 
 the circle, their curvature may be thus 
 obtained from Fig. 22. By construction 
 the points 5 and 7 in the plan are in a 
 line parallel to X X. Through 5 6 and 7 
 draw lines parallel to X X, producing the 
 points 5 and 6 into the arc 5E 6 and the 
 points 6 and 7 into the arc 6 B 7. Draw 
 6 9 perpendicular to x x to 9 in 57. 
 Then 6 9 is equal to the chords 6 7 and 
 6 5 of the circular arc 6B7 and 5E6. 
 Produce the chords 6 7 and 5 6 beyond 
 this arc, and make A c and c C equal to 
 6 X or X 7, and make D d and d E equal 
 to 6 or 5. Then ABC will be the 
 circular curve of the heading joints 67 
 6' 7', and D E F will be that of the cours- 
 
65 
 
 ing joint 5 6 5' 6'. Their middle ordi- 
 nates are the same as those of the arcs 
 6 B 7 and 5 E 6, or both equal to B C or 
 ED. It should be noted that while 
 ABC and DEF are approximate values 
 of, the chords AcC, and DdE are the 
 actual chords of the true heading and 
 coursing joint curves. 
 
 Fig. 33 
 
 54. The radii of these circular approx- 
 
66 
 
 imate curves are thus obtained. Let p 
 be the length of each of the arcs 6 B 7 
 and 6 E 5 or 69 in the development ; I 
 their chords or 6 9 in the plan ; b the 
 breadth 5' 6', and w the width & T of 
 the soffit development, c the chord A c C, 
 and c chord D d F ; m the middle ordi- 
 nate B c or E d ; E the radius of A B C> 
 and E' that of DEF. Then for the 
 given width w the axis X'X' will pass 
 through the middle points of w and #, 
 
 ?/> 
 
 when b= r-^a ( 19 ) 
 
 tang, ft 
 
 or p=w cos. /3=b sin. /3 (20) 
 
 whence we have 
 
 1=2 r sin. 1-2- . 180\ = 2r sin Y . 90) 
 \7rr / \7tr / 
 
 (21) 
 (22) 
 
 in the^triangle 679, 67 c, 69 =Z, and 
 79=7'9'=^ sin. /?, 
 
 or <? a =/ 2 + (^sin. /?) 2 (23) 
 
67 
 
 but -^-=2K-m .-. K=| Jro (24) 
 
 ra 8w 
 
 and in the triangle 569, 56 =c', 6 9 = J, 
 and 59= cos. /?, 
 
 os. /?) 2 (25) 
 
 or 
 and 
 
 Sw 
 
 (26) 
 
 55. Fig. 23 is the side view and plan 
 of the templet for the soffit face of the 
 voussoir, or simply of the soffit face 
 templet. The blade A B C is of the ex- 
 act length and curvature of the soffit 
 heading joint of the voussoir, and is per- 
 
68 
 
 manently attached to the stock, so that 
 its edge C D is normal to the curve ABC 
 at C, and blade E F G is also normal to 
 the stock CD at C, and its curved edge 
 G F E is the curve of the soffit coursing 
 joint. 
 
 56. The soffit face templet is thus ap- 
 plied. After the first coursing bed has 
 been worked as described in article 50, 
 apply the stock to it and work off the soffit 
 face of the stone until the curve E F G 
 coincides with the soffit coursing joint 
 line already worked on the coursing bed, 
 at the same time the curve A B C is ap- 
 plied, and the voussoir soffit surface will 
 be worked throughout its entire length 
 to the curve A, B C. Then on the soffit 
 face so worked, lay off a line parallel to its 
 coursing joint, already obtained, and dis- 
 tant from it a distance equal to the curve 
 ABC of the soffit face templet for the 
 second coursing joint. Then the soffit 
 face of the voussoir is determined. 
 
 57. The soffit face templet is then re- 
 versed. Its curve A B C is applied to 
 the warped soffit and its curve E F G to 
 
69 
 
 the second coursing joint just obtained, 
 and the second coursing bed is worked to 
 the stock C D. Thus let 1 2 3 4 (Fig. 24) 
 be the end view of a voussoir whose upper 
 coursing bed 1 2 and soffit face 2 3 has 
 been worked, the soffit face templet is 
 shown applied to work the second cours- 
 ing bed 3 4. The extradosal coursing 
 joint on this second bed can be obtained 
 in the same manner it was on the first 
 bed, and then the back of the stone can 
 be worked off, and the whole voussoir 
 will be worked. 
 
 ANOTHER METHOD OF WORKING THE VOUSSOIRS. 
 
 58. First work in one bed of the stone 
 a convenient portion of the soffit cylin- 
 drical surface. This may be done by 
 cutting into it the drafts of the arcs of 
 two right sections of the soffit, and then 
 working the face to these drafts and on 
 lines normal to these sections. This may 
 be done with the templet shown in Fig. 
 25. The two arms a abb and decf are 
 permanently secured to one another so 
 that the parallel edges a a and b b and 
 
70 
 
 h 
 
 e 
 
 8 
 
 Wl 
 
71 
 
 the plane of the arm a a b b are at right 
 angles to the plane of the arm d c ef. 
 The lower edges of the two arms coin- 
 cide at their point of intersection, and 
 the curve e cf is a circular arc of the sof- 
 fit right section. 
 
 A line drawn on a cylindrical surface 
 normal to the right section of this sur- 
 face is a straight line parallel with axis, 
 or in other words, it is an element of the 
 cylindrical surface. If, therefore, two 
 drafts be cut in the face of the stone to 
 receive the lower edges of this templet, 
 the straight draft will be an element of 
 the cylindrical surface of which the 
 curved draft is a right section, and if 
 either arm be moved over its draft with- 
 out departing from the plane of this arm, 
 and the stone be worked off to receive the 
 sweep of the other arm of the templet, 
 the surface thus obtained will be the soffit 
 cylindrical surface required. If the circular 
 arm extends sufficiently on either side of 
 the straight edge to sweep the entire soffit 
 of a voussoir, the templet may be con. 
 fined to the single movement of the arm 
 
72 
 
 a abb over its straight draft, and this 
 arm may be maintained within a single 
 plane in this movement by a spirit bubble 
 on the upper edge of d e c f. In working 
 this cylindrical surface care should be 
 taken that the straight draft should be 
 cut deep enough that the sweep of the 
 circular arm shall be entirely within the 
 body of the stone. Otherwise the vous- 
 soir soffit may be deficient, and the pro- 
 cess of working the soffit will have to be 
 repeated on new drafts cut deeper into 
 the stone to receive the templet arms. 
 
 Through any convenient point A, 
 Fig. 26, of a cylindrical surface thus 
 worked draw an element A E. If 1 2 be 
 the straight draft line over which the 
 arm a a b b of the templet (Fig. 25) moved 
 to determine this surface, AE is deter- 
 mined by drawing a line through A paral- 
 lel to 1 2. A E may be determined by 
 applying the soffit templet to and draw- 
 ing on the worked soffit the line A E 
 along the edge a a of the templet, the 
 edge a a passing through the point A in 
 the worked soffit. 
 
73 
 
 Let A' B' C' D', Fig. 27, be the soffit 
 development of any voussoir with the 
 line A' E' drawn on it parallel to the axis 
 or spring lines of the arch. Now cut a' 
 templet of card, lead, zinc, sheet iron 
 
 Fig. 36 
 
 or other flexible material exactly to 
 its pattern, with any convenient hole a 
 cut through it to show the coincidence 
 of the line A E, Fig. 26, with A' E' when 
 
74 
 
 this templet is applied to the worked soffit. 
 Then apply A' and line A' E' of the temp- 
 let to A and A E, Fig. 26. Curve the 
 templet exactly to coincide throughout 
 with the worked surface (Fig. 26), and 
 trace on the soffit the edges of the vous. 
 soir as shown by dotted lines. 
 
 59. The templet, shown in Fig. 28, is 
 the same as Fig. 25, except the spirit 
 
75 
 
 level is removed and the arm fg is added 
 to it. This arm is permanently attached 
 to and is in the same plane as the circular 
 arm de of, and its inner edge/*/ is nor- 
 mal to the circular arc e cf. The cours- 
 ing beds of the voussoirs are worked to 
 this templet. The arms a abb and decf 
 are applied to the soffit, care being always 
 taken that the edge a a always coincides 
 with an element of the cylindrical sur- 
 face. In working the bed at A D the 
 templet is applied to the soffit and the 
 point/* is moved along the joint AD and 
 the stone below is worked off to receive 
 the &rmfg. To work the bed below B C 
 the templet is reversed, and the point / 
 then moves along B C and the coursing 
 bed below B C is worked off to receive 
 the arm f g. The ed ge f g, in application, 
 is normal to the heading joints and to 
 the coursing joints of the soffit when the 
 point/* is at A, B C and D. The linefg at 
 such points is therefore at the intersection 
 of the heading surfaces and of the cours- 
 ing beds when so applied. If, therefore, 
 the templet be applied at A and the di- 
 
76 
 
 rection of fg marked on the coursing bed 
 below AD and then at B and the direc- 
 tion of fg worked on the bed below B C, 
 the heading surface is determined and 
 can be worked to the two lines thus de- 
 termined and to the heading joint A B. 
 In like manner work off the heading sur- 
 face at C D. 
 
 60. By applying the spirit level, Fig. 
 25, to the coursing bed templet, Fig. 28, 
 and a thumb screw at /*, so that the arm 
 fg can be removed from or attached to 
 the blade de cf^ we would have the 
 soffit and coursing bed templets com- 
 bined in a single templet. Economy will 
 sometimes render such a templet desir- 
 able. It will, however, in the construc- 
 tion of oblique arches of any magnitude, 
 be found best to use the two separate 
 distinct templets here shown. 
 
 61. This process has the great advantage 
 of determining all the joints and surfaces 
 of a voussoir with great exactness. It, 
 however, requires great care in applica- 
 tion, simple as is its theory. Care should 
 be exercised, 1st, in working the soffit to 
 
77 
 
 ii 
 
 V 
 
 00 
 
 fci 
 
 true cylindrical surfaces ; and 2d, in work- 
 ing the coursing beds for the soffit, that 
 the templet be so applied that the lower 
 edge of the straight edge arm always co- 
 incides with an element of the soffit. 
 
78 
 
 INCREASED COST OF WARPED SURFACES. 
 
 62. No exact rules can be given for the 
 percentage of the waste of material in 
 working voussoirs to helicoidal warped 
 surfaces over that of working them to 
 straight surfaces in right arches. For 
 besides the diameter and obliquity of the 
 arches this percentage is dependent upon 
 the three dimensions of the voussoirs. 
 
 If we consider an oblique arch of ordi- 
 nary diameter, but of an unusually sharp 
 angle of obliquity, the per cent, of this 
 waste and cost of working will, of course, 
 be unusual. Let an oblique arch of 30 
 feet diameter, and of 40 obliquity be 
 such a case, and we will suppose the arch 
 to be 2.50 feet deep. = 50. The right 
 semi-circumference or 7rr=47.17 and from 
 eqs. (11), (13) and (14)/?=37 -llJ'/?' = 
 41-31' and G S, Fig. 16, 59.165. Let 
 45 be the convenient number of voussoirs 
 for the face of the arch, then the length 
 
 of the bedding joint will be = 1.315 
 
 whence for Eq. (24) E=23.07. If 3 feet 
 is a convenient soffit length for the vous- 
 
79 
 
 soirs their extradosal length will, Eq. 
 (25), be 3.61. Eq. (6) gives # = 37- 
 11J' or equal to ft and Eqs. (4) and (5) 
 makes the extreme warp to 0.25 or 3".* 
 The middle ordinate of the soffit heading 
 joint is .009, and that of soffit coursing 
 joint is .012. The intradosal width is 
 1.31 and the extradosal 1.45. 
 
 63. Let Fig. 29 be an end view of a 
 
 * According to Buck this extreme warp = 3 x tan. 
 (j8'-j3) = 0.227, or a full quarter of an inch less than the 
 actual warp. (See art 20). 
 
80 
 
 helicoidal voussoir. The waste of mate- 
 rial will be a wedge C DE or B A F whose 
 width is the length of the voussoir, or if 
 the voussoir be of the warp and dimen- 
 sions given in Article 56. This waste 
 will be 1.03 cubic feet. The contents of 
 the warped voussoir or its equivalent 
 straight voussoir are 11.33 cubic feet, or 
 the waste of material is less than 8J per 
 cent., since the contents of the block 
 from which the warped voussoir curve is 
 12.36 cubic feet. 
 
 Allowing $1.60 @ perch of 16 feet for 
 quarry stone the block for the straight, 
 and warped voussoirs will co'st respect- 
 ively, 
 
 Straight. Warped. 
 
 dh-j -I O db-| f\A 
 
 p 1 . 1 tjp 1 . 4>t 
 
 There are 32j square feet dress 
 
 surface in each case, which @, 
 
 12c. is 3.90 3.90 
 
 Extra on account of the warp ... 50 
 
 $5.03 $5.64 
 
 Or the warped voussoirs will cost 12 
 per cent, more than the straight ones. 
 This per cent, in practice would be less 
 
81 
 
 than 10 per cent. This comparison has 
 been made on the supposition that the 
 block stones were quarried exactly to the 
 given dimensions in both cases. In 
 practice, however, they are quarried con- 
 siderably larger than needed. The block 
 stone would probably be of the same 
 dimensions in both cases. Again, our 
 illustration is an unusual one. In an 
 ordinary obique arch the obliquity is 
 much less, and the per cent, of cost would 
 be less than 10 per cent. If the work- 
 men are skilled in the use of the templet 
 they will cut warped voussoirs as rapidly 
 as straight ones, and the skill is soon 
 acquired. The extra cost for templets is 
 insignificant, in work of any magnitude. 
 
 SKEW-BACK VOUSSQIR. 
 
 64. It will be observed by reference to 
 Fig. 17 that the intradosal development 
 of the voussoirs at the spring lines S U 
 and G Q are right angled triangles. Let 
 Fig. 30 be an enlarged cut of one of 
 these triangles. Its spring line length I 
 or C B is obtained by dividing its whole 
 
83 
 
 spring line G Q by the number of these 
 triangles upon it. Draw p or A C per- 
 pendicular to /, and let the two parts of 
 I thus divided be t and t' . The angle 
 ABC will be equal to ft the angle of 
 the intrados, whence we obtain 
 
 w=l sin. ft (27) b = l cos. ft (28) 
 pw cos. ft b sin. ft (29) 
 
 t=w sin. ft (30) and t' = b cos. ft (31) 
 
 The triangular extradosal development 
 of these voussoirs are shown in Fig. 18. 
 Any of these spring lines, I being an ele- 
 ment of the intrados parallel to the axis 
 of the arch is a straight line, and the 
 corresponding spring line of the extrados 
 is equal to and parallel to I. Their im- 
 post surfaces are plane surfaces. If they 
 are constructed without being made part 
 of the course immediately below them, 
 they will crack off at B and C, Fig. 30, 
 and the tendency to move over the im- 
 post will be great. But if they are made 
 part of the course below this weakness 
 will be overcome, and the tendency to 
 
84 
 
 move over the impost will be abutted by 
 the wing walls of the arch. 
 
 65. These skew-back voussoirs ar- 
 ranged as suggested by Article 64, are 
 thus constructed. The coursing bed and 
 soffit joint at A B, Fig. 30, is worked for 
 its length AB the same manner as that of 
 any voussoir in the body of the arch is. 
 Let Fig. 31 be an end view of one of 
 these skew-backs, A C G H being that of 
 the heading surface. The line A H is 
 normal to the soffit coursing joint of the 
 bed just worked, and is therefore deter- 
 mined. 
 
 Then apply the stock of the soffit face 
 templet to A H and determine C, Figs. 
 30 and 32, Fig. 32 being a perspective 
 view of the worked skew-back. Work a 
 straight line C B and let the soffit face 
 templet be applied and the soffit between 
 A B and B C worked. Lay off on CB 
 CDt, Eq. (30), and apply the curve 
 A D of the templet A D E, Fig. 31, to the 
 A D, Fig. 32, and work the line D E in 
 the face of the skew-back. In the templet 
 A D E,A D is the curve of the right section 
 
85 
 
 of the arch and the straight edge D E 
 departs from a normal to this curve at D 
 in conformity to the batter of the face of 
 the abutment. The line B I, Fig. 32, is 
 normal to the curve A B and is deter- 
 mined in working the bed A B I H. 
 Therefore, make C G normal to C B and 
 
86 
 
87 
 
 parallel to I B, and work the heading 
 surface A C G H. The face of the skew- 
 back is then worked off on lines parallel 
 to D E, and the bed of the skew-back par- 
 allel to the impost plane B C G I. When 
 the skew-backs are thus cut the stone 
 from whicht hey are worked may be quar- 
 ried without waste of material over that 
 of the voussoir in the body of the arch. 
 
 CHAPTER VI. 
 
 METHOD OF WORKING THE RINGSTONES, CEN- 
 TERING, &C. 
 
 Ring Stones. 
 
 66. Let Fig. 32 be the development of 
 an arch face end of the intrados and the 
 extrados between the axis X'X' and their 
 spring lines S V and R T showing the 
 end curves X'S and X'R and the succes- 
 sive coursing joints of each surface. 
 These end curves should be exactly de- 
 termined by Eqs. (17) and (18) and the 
 coursing joints by Articles 45 and 46. 
 
 Beginning at the axis X'X' let the cor- 
 
88 
 
 responding intradosal and extradosal sur- 
 faces of the successive courses be num- 
 bered 1, 2, 3, 4, &c., and their correspond- 
 ing coursing joints, 1, 2, 3, 4, &c., as 
 shown. 
 
 To determine the angle an end line 
 c a of the intrados between any two 
 coursing joints 3 and 4 makes with these 
 joints, through a, the point of intersec- 
 tion of the joint 4 with the curve 'X'S, 
 draw a b perpendicular to the joint 3. 
 Then if A B C D, Fig. 34, be the soffit of 
 a properly worked voussoir, at any con- 
 venient point d draw de normal to C D 
 or joint 4, which may be readily done 
 with the soffit face templet, Fig. 34. Lay 
 off e f on A B or joint 3 equal to b c, Fig. 
 33. If then a flexible straight edge be 
 applied to the soffit and d f drawn upon 
 it, d f will be the proper location of the 
 end line ac, Fig. 34. 
 
 All other arch face lines of the soffit 
 may be in a like manner determined. 
 
 67. To determine the angle the joint 
 of any coursing bed 5 5, in the face 
 of the arch makes with its soffit coursing 
 
U-- - 1 -- 
 
90 
 
 joint, through h the joint of intersection 
 of the in tr ado sal joint 5 and the end 
 curve X'S, draw h i in a direction perpen- 
 dicular to the axis X'X', and intersect the 
 extradosal joint 5 produced at i. By con- 
 struction h and i are the intradosal and 
 extradosal extremities of the element of 
 the warped coursing bed. Therefore i j 
 is the distance on the extradosal joint 5 
 that the point j of the intersection of the 
 extradosal joint 5 and X'B, is from the 
 extradosal extremity of the normal line 
 to the soffit coursing joint 5 passing 
 through h. 
 
 Then if AB and CD, Fig. 35, are re- 
 spectively the intradosal and extradosal 
 joint 5 5 of the coursing bed A BCD of 
 a properly worked voussoir, apply' the 
 soffit face templet to it at any convenient 
 point d, and determine the normal da 
 to the joint A B at d. Lay off on C D, 
 a b equal to ij, Fig. 29, and applying the 
 flexible rule to the coursing bed, draw 
 d b from d to b, for the proper location 
 of the arch face joint of the coursing 
 bed 5 5, and for the courses 4 and 5. 
 
91 
 
 In this and the preceding article the 
 entire half of the two curved ends and 
 their coursing joints have been consid- 
 ered, for the convenience of illustration. 
 
 Fig. 34 
 
 A / * 
 
 . Fig. 35 
 
 n e J c 
 
 Such consideration is not always con- 
 venient to entertain, but the parts relat- 
 ing to one or more courses may be treated 
 separately by methods that readily sug- 
 gest themselves. 
 
92 
 
 68. True arch face soffit and bed joints 
 must of course all be within a single 
 plane surface, since the arch face is a 
 plane surface. Hence it is that the 
 methods given in articles 66 and 67 are 
 approximate. Their resultants, however, 
 are practical solutions of the true arch 
 face joints. Their true and exact curves 
 may be determined in the following man- 
 ner: Let Fig. 34 be the soffit of course 
 3, and Fig. 35 the coursing bed between 
 courses 3 and 4, and let the points d and 
 d in Figs. 34 and 35 be identical. Then 
 determine b and /as before, b, d and / 
 are in the plane of the arch face. The 
 voussoir is then worked to a plane sur- 
 face passing through the points b, d and 
 f. This method is true and requires but 
 little more time and care to execute than 
 the methods of articles 66 and 67 do. 
 
 CENTEKING. 
 
 69. The centering ribs should be 
 placed in planes parallel to the face of 
 the arch, and, therefore, when so ar- 
 ranged will be elliptical. They are some- 
 
93 
 
 times placed normal to the soffit, or made 
 circular, when, in order to receive the 
 arch under its acute angle, the centering 
 has to be extended beyond the obtuse 
 angle, and there loaded to prevent any 
 movement in the centering when the 
 voussoirs are set near the actual angle. 
 Circular centering should not be employed 
 to receive the voussoirs of an oblique 
 arch. The ribs of the elliptical centering 
 being parallel to the arch face, are in the 
 planes of pressure, are easily maintained 
 and require no more material than is nec- 
 essary to receive the voussoirs. 
 
 70. The sheeting or lagging should be 
 so put on the ribs that it will have an 
 even and smooth surface, and that the 
 centering will be of the exact dimensions 
 to receive the voussoirs. When so pre- 
 pared the soffit coursing joints of every 
 course should be carefully and perma- 
 nently marked upon the sheeting, as a 
 guide for the placement of the voussoirs 
 in the arch. The location of these joints 
 on the centering may be determined from 
 the development of the intrados. As the 
 
94' 
 
 skew-back stones are generally set before 
 the centering is, no lagging or sheeting 
 will be required on the centering below 
 the intradosal upper courses of the skew- 
 back stones. 
 
 SEGMENTAL AND ELLIPTICAL ARCHES. 
 
 71. As the same principles are involved 
 in segmental as in full semi-circular right 
 section of a helicoidal oblique arch, no 
 further rules are necessary for their con- 
 sideration. 
 
 As in right segmental arches the axes 
 of oblique segmental arches are in the 
 planes of their imposts or springing sur- 
 faces, and their skew-back stones should 
 be constructed accordingly. 
 
 Elliptical oblique arches are not rec- 
 ommended, both on account of their 
 structural weakness and the difficulties 
 involved in their construction. 
 
 THKUST OF THE ARCH. 
 
 72. The thrust of a helicoidal oblique 
 arch being carried to the impost in lines 
 parallel to the face of the arch causes a 
 
95 
 
 tendency in the 'arch to move outward at 
 the acute angle of the arch, which is re- 
 sisted by the friction of the coursing 
 beds of the voussoirs. This tendency to 
 move increases with the acuteness of the 
 angle of obliquity, and when very acute 
 this tendency should be resisted by prop- 
 erly constructed wing walls. When so 
 constructed these arches may be con- 
 structed to any desired angle. It will, 
 however, be a rare case where the angle 
 of obliquity is less than 25 the limit 
 calculated by John Watson Buck. If he 
 had considered the arch helicoids con- 
 structed as recommended in article 44 the 
 construction would have been more stable, 
 and his limit less than 25.* 
 
 * Helicoidal oblique arches are much more stable 
 than they are generally supposed to be. During 
 the summer of 1877 the author superintended the con- 
 struction of two of these arches, of 66 feet cylindrical 
 length, 16 feet right diameter, and of 40 obliquity, 
 each, and though placed under two tracks over which 
 the heaviest railroad traffic passed, these arches, at 
 this time, March, 1886, do not evince the least weak- 
 ness. No extra precaution was taken to prevent the 
 skew-back stones moving over their beds. A right 
 arch could not have performed the work better or 
 more satisfactorily. 
 
96 
 
 Logarithmic and Ribbed Oblique Arches, 
 
 i. 
 
 LOGAEITHMIC METHOD. 
 
 73. This method of constructing ob- 
 lique arches is so-called because Naperian 
 logarithms are used in their calculations. 
 
 The soffit coursing joints by this meth- 
 od are always normal to the plane of the 
 arch face, wherever they come in contact 
 with it, and hence it is these coursing 
 joints are normal to any plane parallel 
 with the arch faces at their points of con* 
 tact in the parallel planes. The soffit 
 heading joints are elliptical curves in 
 planes parallel with the arch faces, and 
 are, therefore, normal to the coursing 
 joints of the soffit at their intersections. 
 The heading and coursing joints of the 
 soffit being thus normal to one another, 
 they will also be normal to one another 
 in the developed soffit. 
 
 74. Fig. 36 shows the plan and devel- 
 
97 
 
 opment of the soffit of a semi-circular 
 oblique arch, whose elevation and right 
 section is shown at ABC and H IJ. 
 
 The curve N X'K normal to the curve 
 S X'G at its middle joint X' is the de- 
 
98 
 
 veloped soffit coursing joint through that 
 point. Through the middle point R of 
 the spring line S O, draw the dotted line 
 R P parallel to the curved ends of the 
 soffit SG and O Q. Divide RP thus 
 drawn in to any convenient number of 
 parts of equal length (Fig. 37) and 
 through the point of division, draw their 
 coursing joints parallel to the curve 
 N X'R. The widths of the courses are 
 thus determined on the middle curve 
 R P, in order to show the same order of 
 arrangement and of size of the several 
 courses in each arch face, but their dimen- 
 sions may be fixed on any other parallel 
 curve to R P. It will, however, be found 
 most convenient to take the middle curve 
 R P, and also to make the courses of one 
 width on this curve. 
 
 75. Having thus determined the posi- 
 tion of the coursing joints in develop- 
 ment, the heading joints are drawn 
 in the several courses at desired or 
 convenient points, and their elliptical 
 curves are drawn parallel to R P, or to 
 the curves of the developed arch ends. 
 
99 
 
 ig. 37 
 
100 
 
 It should be borne in mind that the 
 heading and coursing joints in the devel- 
 opment are drawn parallel with E P and 
 with N X'E on lines parallel to the spring 
 lines S O and G Q. Thus, if we cut out 
 a cardboard templet, one of whose edges 
 will correspond with the curve N X'K 
 and its left-hand edge is straight and 
 parallel to S O, and through the points 
 of division in E P we draw the several 
 coursing joints shown in Fig. 37, this 
 templet should be moved so that its left 
 hand straight edge shall always be par- 
 allel to the spring line S O. In like man- 
 ner the heading joints may be drawn with 
 a templet of curvature S X'G, but in mov- 
 ing it over the development S O Q G, its 
 extremities S and G should always move 
 on the spring lines S O and G Q, whilst 
 the curved face of the templet moves 
 parallel to the end curves S G and O Q. 
 These precautions should always be ob- 
 served, otherwise the curved lines drawn 
 would not be correct or in accordance 
 with the requirement of article 69. 
 76 The equation of the normal curve 
 
101 
 
 N X'K to S X'G at its middle point X' 
 will now be determined. 
 
 In article 41 will be found the expres- 
 sion 
 
 2/ r sin. a tang. 6 (32) 
 
 Now since x is dependent upon a for 
 its value this equation contains a ratio of 
 x to y, and is therefore an equation of 
 the end curve S X'G, and might be used 
 for it in place of equation (17). Let# be 
 the complement of a, and we have 
 
 yr cos. b tang. 6 (33) 
 
 Again a is usually expressed as so 
 many degrees, or as W. 180. It is in 
 fact the length of an arc of n 180 to 
 radius unity. Thus, the expression tang. 
 36 means the tangent to an arc of 36 
 to radius unity, but not the tangent of 
 36. a therefore equals n TT, 
 
 but x=H7rr, .: x=r a (34) 
 but a-^ b y .'. x=-~ rb (35) 
 
 A A 
 
 Differentiating equations (33) and (35) 
 and dividing we have : 
 
102 
 
 du r tan. sin. b db 
 
 -7-= tan. u sin. b 
 
 ax r db 
 
 (36) 
 
 Now, let y (Fig. 36) be the ordinate to any 
 point in N X'K that y is for the correspond- 
 ing point in S X'G for the ordinate x. At 
 an infinitesimal distance from the point 
 of contact X' the curves N X' K and 
 S X'G are straight lines. . Thus, in Fig. 
 38, let X' be the origin of co-ordinates 
 XT the axis of y, and X'X that of x. 
 Now, if on X X' we lay off an infinitesi- 
 mal distance X'B, and through B draw 
 A C parallel to X'Y, the curves AX' and 
 C X' are exactly at right angles to one 
 another within these limits, and the ordi- 
 nates to A will be : 
 
 dx and dy r and the ordinates to C will 
 be 
 
 dy and dx. The triangles AX'B and 
 BX'C are similar, whence 
 
 dy : dx : : dx : dy' , 
 
 *Z=+p, (37) 
 
 ' 
 
103 
 
104 
 
 but -=r = tang. 8 sin. b (see eq. 36) 
 
 r/r 
 
 /. =tan. sin. 5(38). Buidx=rdb 
 dy r 
 
 t __ rdb r db 
 
 tan. 6 sin. b~ tan. ' sin. b 
 r db 
 
 tan. 6 2 sin. -3- 5 cos. -J # 
 
 r c?5 cos. 2 
 
 tan. 6 * cot. \ b 
 oi ^ b (39) 
 
 .-. y'=r cot. log. cot. %b + C (40) 
 or, substituting (90 a) for 5 we have: 
 y'=r cot. log. cot. (90 -a) + (41) 
 as the equation of the curve N X'K in 
 which if 
 a=Q, x=Q, and log. cot. (90 a) = 
 
 or 0=0, 
 whence the equation of NX'K becomes 
 
 y'r cot. 6 log. cot. J (90 ) (42) 
 wherein, if a and #=0, y=0 
 
 if a=90, x=-r and = cc 
 
105 
 
 TtT 
 
 if a= 90,cr, = andy=oc 
 
 A 
 
 By the aid of Eqs. (42) and (32) the 
 soffit coursing and heading joints in the 
 development may be determined with 
 great precision. 
 
 77. The coursing beds of oblique arches 
 by the logarithmic method are generated 
 by a radial line normal to, and moving 
 along the axis of the arch as one direc- 
 trix, and on a cylindrical curve as the 
 other directrix, which at all points is nor- 
 mal to planes parallel to the arch faces. 
 
 This second directrix is usually taken 
 in the soffit, and we will so treat it here, 
 but should be in cylindrical surface, 
 midway between the in tr ados and the ex- 
 trados. 
 
 78. There is great similarity in the gen- 
 eration of the coursing bed surfaces by the 
 helicoidal and by the logarithmic meth- 
 ods. It should be noted both are gener- 
 ated by a radial line normal to, and mov- 
 ing along the axis of the arch. Their 
 difference is in the fact that, in one case 
 the second directrix is a helix, and in the 
 
106 
 
 other, a normal curve to the arch faces. 
 It matters little what this second direc- 
 trix is so long as its curvature is known. 
 But it is of the greatest importance that 
 we keep in mind that the first directrix is 
 radial in the logarithmic just the same as 
 it is in the helicoidal method ; nor should 
 this idea, in the treatment of oblique 
 arches by either of these methods, be 
 ever lost sight of. It is the fundamental 
 principle and renders these two methods 
 quite similar in construction, and for this 
 reason the treatment of logarithmic 
 arches is readily understood when the 
 problems connected with helicoidal arches 
 have been once mastered. 
 
 79. Again, it should be noted the only 
 straight line elements in the coursing 
 beds by either methods are radial ; that 
 is, they are in lines normal to the axis of 
 the arch, and consequently, are always 
 normal to the coursing joints, both intra- 
 dosal and extradosal. The only straight 
 lines in the soffit, by either methods, are 
 lines parallel with either the axis or the 
 
107 
 
 spring lines of the arch, nor should this 
 fact be lost sight of. 
 
 80. Now, if NX'K (Fig. 36) is the in- 
 tradosal joint and MX'L the extradosal 
 joint of a coursing bed passing through 
 X' in the development, and if through 
 any point 1 in NX'K we draw 1 2 in direc- 
 tion perpendicular to S,O continued, the 
 point 2 in MX'L is the intersection of 
 the radial element through 1 in NX'K. 
 We have already determined the ordinate 
 of 1 to be 
 
 y ' -r cot. log. cot. 4 (90 -a) (42) 
 but y' is also ordinate of the point 2, 
 and Eq. (42) is therefore the equation of 
 MX'L when 
 
 x'=r'a (43) 
 
 II. 
 
 METHOD OF WORKING THE VOUSSOIB. 
 
 81. Reduce the face of the stone to be 
 worked to a true cylindrical surface by 
 aid of the soffit templet, Fig. 25, in the 
 manner as described in article 58. This 
 reduction may be accomplished in a va- 
 
108 
 
 riety of ways, but the method there de- 
 scribed is believed to be the simplest and 
 best. 
 
 Let A B C D, Fig. 39, be the soffit thus 
 reduced, and let A'B'C'D' be the devel- 
 oped soffit of the voussoir to be worked. 
 Through B with the straight blade of the 
 soffit templet draw the element BE, and 
 through B' in the development draw cor- 
 responding line B'E' parallel to the 
 spring lines, or to the axis of the arch, 
 and through D' draw D'F' parallel to 
 B'E', and through A' and C' draw A'G' 
 and C'F' perpendicular to B'E' and to 
 D'F', respectively. Lay off on B E, B G 
 and GE equal to B'G' and to G'E'. 
 Then with the curved blade of the soffit 
 templet through G and E draw the cir- 
 cular arcs G A and E C, whose respective 
 lengths shall be equal to A'G' and E'C' 
 in the development. 
 
 Produce the arc E C to F and make 
 E F equal to E'F' in the development, 
 and with the straight blade of the soffit 
 templet draw the element F D parallel to 
 B E, and make F D equal to F'D' in the 
 
109 
 
110 
 
 development. The four corners A, B, C 
 and D of the voussoir soffit are now de- 
 termined. 
 
 82. The heading and coursing joints 
 of this soffit are reduced in a simple man- 
 ner. Thus let a flexible rule of card- 
 board, thin hard wood, or other suitable 
 material, be cut, with one edge of the 
 exact length and curvature of the devel- 
 oped joint A'B', and apply its extreme 
 points A' and B' at A and B, press the 
 rule against the cylindrical soffit, and 
 cause it to thoroughly conform to this 
 surface, and whilst so applied, work on 
 the soffit between A and B the line of 
 the curved edge of the rule, and the 
 joint AB will be determined. The joints 
 B C, C D and DA may be thus deter- 
 mined by rules cut to their developed 
 curves, and then applied to the worked 
 soffit of the voussoir in like manner as 
 A B was determined. 
 
 83. Another way of determining the 
 voussoir corners and joints in the worked 
 cylindrical soffit is : Let any corner B, 
 Fig. 39, be selected as before, and 
 
Ill 
 
 through it draw the element B E. Then 
 cut a templet of cardboard, or of other 
 flexible material, to the exact size of the 
 developed soffit A'B'C'D', and draw 
 across it the line B'E' parallel to the 
 spring lines of the arch, and apply this 
 templet with its corner B' at B and its 
 line B'E' on BE, and cause the templet 
 to conform throughout to the worked 
 cylindrical soffit surfaces, and when so 
 applied, work all the edges of the temp- 
 let on the stone, and thus determine all 
 the joints of the soffit at one operation. 
 
 REDUCTION OF THE COUESING BEDS. 
 
 84. When it is remembered that the 
 coursing beds by this method (logarith- 
 mic) are generated by a radial line, it will 
 at once appear that the coursing bed 
 templet, Fig. 28, is as applicable to log- 
 arithmic arches as to helicoidal arches 
 for the reduction of the coursing beds, 
 when the coursing joints have been lo- 
 cated upon the worked soffit of a vous- 
 soir. 
 
 85. The surfaces below the joints A D 
 
112 
 
 and B C, Fig. 39, are therefore worked to 
 the radial edge of this templet. When 
 so worked, radial lines are. drawn with 
 this radial edge on the coursing beds 
 through the corners A, B, C and D, thus 
 determining the lines of intersections of 
 the heading and of the coursing beds of 
 the voussoir. The heading surfaces are 
 then worked to these radial lines so 
 drawn, causing the surfaces to be normal 
 to the soffit, and therefore normal to the 
 coursing beds. 
 
 86. Much has been said as to what 
 should be the character of the heading 
 coursing surfaces, both by the helicoidal 
 and by this method. It has been main- 
 tained that these surfaces, by the loga- 
 rithmic method, should be planes parallel 
 to the arch faces ; that the strength and 
 stability of the arch demanded it, &c. It 
 in fact matters little, whether these sur- 
 faces are parallel to the arch faces or are 
 warped surfaces, normal to the coursing 
 beds. It is, however, the author's opinion 
 that the latter construction is the more 
 
113 
 
114 
 
 stable. It has the advantage, also, of 
 simpler construction. 
 
 87. The arch face stones are to be 
 worked precisely in the same manner as 
 described in articles 66, 67 and 68 for heli- 
 coidal arches. In the one the coursing 
 joints are helicoida], whilst in the other 
 these joints are curves, normal to the 
 arch faces. The same principles are in- 
 volved here as there, and therefore do 
 not require a second demonstration. 
 
 88. The logarithmic method of oblique 
 arch construction is one that requires 
 great care and constant supervision to 
 successfully execute. It should be done 
 in the most systematic manner. Thus it 
 will be noticed that the courses increase 
 one side of the arch while they decrease 
 in thickness on the other side of the arch 
 from one end of the arch to the other. 
 Yet, two courses beginning at opposite 
 ends of the arch at the same height 
 above the spring line are exactly alike in 
 all their dimensions. Their voussoirs 
 may therefore be worked in duplicate, 
 which, in fact, is the proper way to treat 
 
115 
 
 this class of construction. These two 
 courses may be laid out at once, and all 
 their voussoirs worked. There is no 
 reason why all the voussoirs of the entire 
 arch may not be worked out before any 
 of them are set in position in the arch. 
 All the voussoirs of a course should be 
 kept separate and distinct by themselves. 
 Every voussoir should be marked in plain 
 figures or characteristics, indicating at 
 once its course and position in it. 
 
 89. The coursing joints of every course 
 should be marked permanently on the 
 centering, as a guide for the voussoirs as 
 they are set in position. This is best 
 done by transferring to the centering or- 
 dinates taken from the development plan 
 of the arch soffit. 
 
 BIBBED OBLIQUE AECHES. 
 
 90. Oblique arches are sometimes con- 
 structed by placing several narrow ellip- 
 tical arches or ribs, as they are called, to- 
 gether, as shown in Fig. 40. This method 
 is very faulty, and cannot be too severely 
 condemned. There is no bond between 
 
116 
 
 the several ribs, as each rib is separate 
 and distinct in its construction and its 
 position ; the load above the arch is never 
 uniform throughout the whole length of 
 the arch, and on account of this lack of 
 bond in the arch, it will be distorted by 
 its unequal settlement. Again, the outer 
 ribs are constantly being forced outwards 
 by the action of frost upon the material 
 that finds lodgement between their head- 
 ing surfaces. So serious becomes this 
 weakness that the ribs hare to be recon- 
 structed to restore the stability of the 
 arch. True, these ribs are sometimes 
 bonded one to another with iron straps, 
 yet this bungling device is devoid of trans- 
 verse strength and is susceptible of tak- 
 ing up the longitudinal stress only. Such 
 an arch is not to be compared to an arch 
 of bonded courses. Again, if it is prop- 
 erly bonded with iron straps, a ribbed 
 elliptical arch costs more than an oblique 
 arch constructed in accordance with the 
 rules of any one of the recognized meth- 
 ods of proper construction. 
 
117 
 
 Strength of Oblique Arches. 
 
 91. The static problems in masonry 
 arches is to so construct, or so arrange 
 the material composing the arch, that the 
 line of pressure resultant from the 
 weight and load of the arch shall pass 
 through it in planes parallel with the arch 
 faces, to the end that the arch stress 
 shall be uniform throughout the length 
 of the arch, and that the courses shall 
 be relieved, as far as possible, of the 
 tendency to move or slide over one an- 
 other. 
 
 92. Without discussing the problems 
 involved, we will suppose that the bond 
 of the arch is ample to resist all ordinary 
 external forces acting upon the arch, and 
 that the arches now to be considered are 
 so constructed that the line of pressure 
 falls within the middle third of the thick- 
 ness of the arch, and that it is located on 
 
118 
 
 the center line of the depth of the arch, 
 as for. instance on the middle line a a, 
 Fig. 41. 
 
 93. Undoubtedly the disposition of the 
 material composing the arch has much 
 to do in determining the direction of the 
 line of pressures. 
 
119 
 
 In a right arch the coursing beds being 
 parallel with the spring lines or axis of 
 the arch, the line of pressure passes di- 
 rectly to the abutments in lines normal 
 to those beds, and therefore in planes 
 parallel to the arch faces. We have seen 
 (article 24 and Fig. 11) that in an oblique 
 arch with straight beds, there was a 
 tendency for the arch stress to pass to 
 the abutments in lines normal to* such 
 straight beds, and it would undoubtedly 
 do so were it not for the distorted strains 
 in the overhanging oblique ends of the 
 arch. 
 
 Let Fig. 41 be an elliptical arch. With 
 straight radial beds it is evident that the 
 line of pressure will be parallel with the 
 arch faces, and this is true if the arch is 
 a single narrow rib. Now, an oblique 
 arch may be regarded as composed of a 
 number of elliptical ribs arranged as in 
 Fig. 40. These ribs may be so narrow, 
 that they will not break the continuity of 
 the arch cylindrical intrados. The line 
 of pressure in any single rib will be 
 parallel to the arch face, and therefore 
 
120 
 
 the line of pressure for the whole arch 
 must also be parallel with the arch face. 
 The line of pressure would not be altered 
 in this respect if the several ribs were 
 reduced to the exact cylindrical intrados 
 of the arch. 
 
 94. Again, in a helicoidal arch there is 
 always a point on either side of the 
 crown, where the coursing beds are nor- 
 mal to the arch face. 
 
 Now, if the arch be limited by imposts 
 or abutments at these normal beds, the 
 line of pressure will evidently pass 
 through the arch normal to the impost 
 and in lines parallel with the arch face, 
 the same as in a right segmental arch, 
 provided that no bed between the im- 
 posts shall depart from the normal to the 
 arch face, equal to the angle of friction. 
 It will be shown directly that this de- 
 parture, for arches of the least practicable 
 obliquity never amounts to as much as 
 one-half of the angle of friction. If the 
 abutment be lowered below the normal 
 beds the line of pressure will continue 
 parallel to the arch face until a bed is 
 
121 
 
 reached whose departure from the nor- 
 mal to the arch face equals the angle of 
 friction. Even then if the surfaces of 
 this lower bed and those below it resist 
 the tendency to move, the line of press- 
 ure will continue parallel with the arch 
 face. 
 
 95. We conclude therefore that the 
 line of pressure is parallel to the arch 
 faces when the condition of greatest 
 stability in an oblique arch is to be real- 
 ized, and that it should be regarded as 
 parallel to the arch faces in discussing 
 the strength of oblique arch. 
 
 96. Perfect stability is realized when 
 the coursing beds are exactly normal to 
 the line of pressure. In the logarithmic 
 method the generating line in the cours- 
 ing beds is exactly normal to the arch 
 face, and, therefore, practically complies 
 with this condition of perfect stability, 
 and needs no further discussion in this 
 particular. 
 
 97. Let S X G, Fig. 42, be the end de- 
 velopment of the arch for the middle line 
 on the arch face between the intrados and 
 
123 
 
 the exfcrados. S XG will also represent 
 the direction of the line of pressure, 
 since it is parallel with the arch face. 
 Draw the straight line SXG, joining the 
 extremities of the development, and draw 
 the middle line of the coursing beds 
 normal to this line, as heretofore de- 
 scribed. At A and B these lines will be 
 normal also to the line of pressure. X, 
 the middle of the end development, is the 
 point between A and B, of the greatest 
 departure of the middle line of a cours- 
 ing bed from the normal to the line of 
 pressure. For an oblique arch of 25 
 obliquity this departure is only 12, the 
 arch, therefore, is for all practical cases 
 perfectly safe between A and B. At C 
 and D the middle course lines depart 
 from the normal to the line of pressure 
 equal to the angle of friction. And the 
 courses below these points will slide over 
 one another if not otherwise resisted. 
 Between C and S the tendency will be 
 for the courses to move over one another 
 into the arch, and between B and G to 
 move out from the arch. It will not be 
 
124 
 
 safe to construct the arch below D, unless 
 precautions be taken to thoroughly resist 
 the outward tendency of the courses be- 
 tween U and G. In an oblique arch of 
 25 obliquity the point D is 10 above 
 the horizontal on the full semi-circular 
 right section, the angle of friction being 
 taken as 36. The stability of oblique 
 arches proves that the angle of friction 
 for the courses of the arch must far ex- 
 ceed 36, especially so for well cemented 
 masonry. The arch of 25 obliquity 
 would unquestionably be safe for 160 on 
 the right section. At 40 above the hori- 
 zontal the coursing bed in this arch 
 would be normal to the arch face. If the 
 arch below 10 was resisted by the wing 
 walls, or if the courses were doweled to 
 prevent slipping the whole arch could be 
 made stable for a full semicircle on the 
 right section. Such construction is not 
 recommended. Segmental arches are far 
 preferable. It is true, however, that full 
 semi-circular helicoidal arches of slight 
 obliquity have been built, that have given 
 the most satisfactory results. 
 
125 
 
 It should be noted the sliding tend- 
 ency of the courses immediately back in 
 the arch from its face between D and G, 
 is opposed by the abutment of the arch, 
 and therefore cannot move without dis- 
 turbing the abutment. 
 
 That the courses in the arch face below 
 D do not, in carefully constructed oblique 
 arches, move the least outward, is ac- 
 counted for by the fact that they are held 
 tight in the arch by the weight of the 
 spandrel or parapet wall above them; 
 that is to say they are so tightly wedged 
 into the arch as to overcome the outward 
 tendency. 
 
 Fig. 42 is an exact drawing of the end 
 development, an oblique arch of 25 ob- 
 liquity for full semi-circular right sec- 
 tion. 
 
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