1840.] 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



Ill 



Add togelhtr the logarithmic cosines of the given parts, and the sum 

 wilt be the logarithmic cosine 0/ the part required. 



Note. — The reader is supposed to liuve a previous knowledge of 

 the trigonometrical definitions, logarithmic tables, and algebraic nota- 

 tion. 



The general application of the table may be described in words at 

 length in the following manner : — 



Add logct/ier the logarithms of the two given quantities according to 

 their names in the equation, and the sum milt gice the logarithm of the 

 required quantitij according to its 7tame in the particular combination 

 employed, observing always to abate 10 in the index of the resulting 

 logarithm. 



Again, to find the angle A B C ^ B, contained between the hypo- 

 thenuse and perpendicular, we have only to refer to that compartment 

 of the table containing the values of B, and to select the combination 

 which involves the given quantities; in this case it is No. -1, from 

 which we have 



tan. B =^ tan. b cosec c ; 

 an equation which is readily reduced by the general rule given above. 



In reference to the arrangement of the table, it may be remarked 

 that it forms a right angled triangle, the same as the figure under con- 

 sideration, and the squares or compartments containing the values of 

 the several parts, are placed in the same positions with respect to each 

 other as the parts are whose values they contain. Thus, in the figure 

 BAG, the hypothenuse a occurs between the angles B and C; so in 

 the table, the square containing the values of tlie hypothenuse, is 

 placed in a diagonal direction between the squares containing the 

 values of the angles B and C. 



In the figure the perpendicular c occurs between the angle B and 

 the right angle at A ; so in the table, the square containing the values 

 of the perpendicular, occurs between the square containing the values 

 of B, and the blank square for the right ang'e where no vakie enters 



Finally, in the figure, the base b falls between the angle C and the 

 right angle at A : so in the table, the square containing the values of 

 the base, is placed between the blank square for the right angle and 

 the square containing the values of the angle C ; an arrangement 

 which is beautifully adapted for the purpose of a speeily reference. 



The two equations that we have selected from the table, are those 

 which apply to the determination of the bevels for the several voussoirs 

 throughout the whole extent of the arch. The first determines the 

 form of the beds, or the angle contained between the joints in the face 

 of the arch, and the corresponding joints along thesotlit; and the 

 second determines the angles contained between the face of the arch 

 and the beds of the several courses. The application of which we 

 now proceed to illustrate by means of an example. 



Suppose a semicircular arch of 30 feet span, and consisting of .34 

 courses from impost to impost, to be built upon an obliquity of 68 de- 

 grees with the abutments, what are the several bevels required for the 

 construction of the arch stones or voussoirs in eacli of the courses ? 



Since the arch is a semicircle of 30 feet span and consisting of 

 34 courses, that is, 17 courses between the crown of the arch and each 

 of the imposts; it follows, that each voussoir occupies 5^ 17' 3b" -i-4 of 

 the circumference, having a soffit or intradosof 2-795 feet very nearly ; 

 consequently, the successive portions of the circumference, estimated 

 from the impost to each of the beds or coursing joints as far as the 

 crown or middle of the keystone, are respectively as in the following 

 tablet. 



The 17th course, or course at the crown of the arch, corresponding 

 an angle of 90 degrees as it ought tu do, when the keystone is in 



two parts, as we have assumed it to be in the present instance, for the 

 express purpose of showing the influence of the obliquity upon the 

 bevels in that course. From these angles therefore, witli the constant 

 obliquity of OS degrees, we derive the following construction for the 

 case in question. 



Let A E B in the subjoined 

 drawing, represent the eleva- 

 tion of the given semicircular 

 arch, of which C is the centre, 

 and A B the span or diameter. 

 At the centre C, make the an- 

 gle A C D equal to 08 degrees 

 the given obliquity, so that C D 

 shall coincide with the axis of 

 the arch, and point out the di- 

 rection of the abutments to 

 which it is parallel. From the 

 beginning of the arch at A, set 

 oft' successively the values of 

 several arcs in the tables cor- 

 responding to the respective 

 number of courses estimated 

 from the impost at A to the 

 crown of the arch at E, and from 

 thtnce in a retrograde order to 

 the other impost at B. 



Upon A C the radius of the 

 arch describe the semicircle 

 A a 6 c C, intersecting the radii 

 C 4, C S and C 12 respectively in the points a, b and c, and at A erect 

 the perpendicular AD meeting C D the axis of the arch in D. 



About A as a centre, with the distances A a, A 6 and A c, describe 

 the arcs af, b e and e d, meeting the radius A C in the points/, e and d 

 respectively, and draw the straight lines D/, D e and Drf: then are 

 the angles A/ D, Ac D and ArfDor their supplements, the angles 

 contained between the face of the arch and the planes of the coursing 

 joints at the 4th, Sth and 12th courses, or at the corresponding divisions 

 on the opposite side of the arch. These are the angles corresponding 

 to the letter B in the figure of the table of formula, and if they are 

 respectively taken in the compasses and applied to a scale of chords, 

 they will be found to indicates I"' 41' 40", 74"^ 46' 2o" and 70° 0' 59". 



Upon the straight line C D as a diameter describe the semicircle 

 C ghiD, in which lay oil" the distances C i,Ch and C g respectively 

 equal to C a, C b and C c ; then will the angles DCk,DCl and D C in, 

 or their supplements, be the bevels of the beds or coursing joints at the 

 4th, Sth and 12th divisions, or at the corresponding divisions on the 

 opposite side of the arch ; the bevels in the two cases being constantly 

 the supplements of each other. 



The angles just determined from the last step of the construction, 

 are those which are measured by the arc a in the tabular figure, and if 

 they are severally taken in the compasses and applied to a scale of 

 chords, they will be found to indicate 69= 33' 17", 73° 55' 12 " and 

 80° 23' 16" respectively, for the bevels in the beds or coursing joints 

 corresponding to the 4th, Sth and 12tli divisions of the arcb. 



The values of B, or the bevels between the face of the arch and the 

 planes of the coursing joints at the specified divisions of the arcb, are 

 also determined from the 4th formula in that compartment of the 

 table containing the values of B. Thus we have tan. B = tan. b cosec 

 c, and taking the parts of the circumference at the respective divisions, 

 we get as follows : 



4th division 21 10 35 ^ - - . log. cosec. 0-442204 



Value of 6 == 68 constant obliquity log. tan. 0-393590 



Value of B= 81 41 40 



log. tan. 0-835794 



Sth division 42 2110-1-? - - - log. cosec. 0-171538 

 Value of 6 = 08 constant obliquity log. tan. 0.393590 



Value of B = 74 46 25 



log. tan. 0.565128 



12th division 63 31 45 -J, - - - log. cosec. 0-048098 

 Value of 6 = OS constant obliquity log. tan. 0-393590 



Value of B = 70 59 - - - - log. tan. 0-441688 

 For the values of a, or the bevels in the planes of the beds or course 



Q2 



