230 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[Aug t, 



continue tn po on thus tlie nineteentli century will be positively a 

 IjImuU in tlic liistiirv (if nrchitccture, and, unlike that of Eliziiheth, 

 till' reif.ni of Victoria an absolute nullity. AVith No. 12(11, "New 

 Huihlini^'s in the Temjile," S. Smirkf, we are by no means so well 

 satisfied. Not onlv are they more continental than Eniflish m 

 physiofrnomy. hut they do not at all accord vrith any thiiitr near 

 them ; on tlie contrary, are ipiite a patch stuck u]ion that range 

 of buildinfrs wliich tjiey are inteiuled to carry on and complete. 

 While others are effaciner and defacing Snane's works, Mr. Smirke 

 seems to he anxious to undo or else go quite counter to Sir Robert s 

 doings. 



ON TRIGONOMETRICAL FORMULiE FOR FINDING 

 THE LEVEL OF TWO DISTANT OBJECTS. 



By R. G. Clark, C.E. 



The object of this communication is to exhibit some simple for- 

 mula", that may at one operation serve to determine the levels of 

 distant ohjects' witli respect to the station from wliich the angles 

 are taken. The subject may be considered thus: we have on a 

 vertical plane II ADB, the given angles H A C, HAD, from HA 

 the sensible horizon, and the given height C D, to determine the 

 level of 1) at the base with respect to B, the base of the station A. 

 (.«('c fig. 1.) 



There are two cases : first, when the vertex A of the station is 

 above the level of the summit of the object CD; and, secondly, 

 when the vertex A is below the level of ('. 



1st. Let the given angle of depression H A C =: ; also the angle 

 H A D = fl ; the given height of the object C D = A ; and H D =: y. 



Then by the triangle CAD we have 



Jl (•QS 



sin (9—$) : ft :: sin (90° — B) or cos s: - , '— ^ = A C. 

 ^ ' ^ ' sin(fl — ;3) 



By the right-angled triangle A H C, we have 



sin -.y — h-.W. '^^^^ = A C. 

 sm/3 



FIg.l. 

 Equating the above values of A C, 



h . cos e 



y- 



(e sin — 0) sin/3' 



Therefore, A . cos 8 . sin e = {y — ft) .sin (9 — /3) 

 = (y — K) (sin 9. cos /3 — cos 6. sin 6) 

 = 2/ . sin 9 . cos 5 — y cos 9 . sin 6 — ft . sin e . cos /3 + ft . 

 sin j3 ; by transposing, j/( sin 6 . cos ;3 — cos 6 . sin /3) = ft 



cos fl 



cos e 



sin e . 

 1. 



T. '* • sin fl . cos 5 . , „N /„N 



Hence « := - -. — ,- „r , or u = ft . sin fl . cos fi . cosec . ( e — &) (2) 



^ sin (fl — /3) ' " ' ^ ' 



Dividing each side of (1) by sin fl. cos S, we have 



y = :; ^ „ . ; hut tan /3 =: ; therefore, 



1 — cot 9 . tan /3 ' cot )3 



ft . cot )5 



^^cotfl — cot^ (-^O 



The formula (2) may be adapted for logarithmic computation 

 thus — ' 



I/OSy = log sin 8 + log cos 6 + log cosec (9 — 6) -f logh— 30. 



The equation ^3) can be effected by natural co-tangents to he 

 found in Huttons " Mathematical Tables." An example is sub- 

 joined to elucidate the equation (2). 



Ex. It being required to determine the level of two objects by 

 angles of depression from the sensible horizon, taken at the summit 

 of an edifice, as St. Paul's, the height being 4.01. (.««■ fijr i ) • tl 

 H A C = 3" 20' ; the angle H A D = 8" 30' ; and the lieia 



= 300 feet 



Here 9 = 8° 30' ; ;3 = 3° 20' ; 



le angle 

 ht C D 



By formula (2) we have 



Log sin 8" 30' 9-169702 



Log cos . 3° 20' 9-999265 



Log cosec . 5' 10 ... 11-0+.5J01 

 Log 300 2-477121 



2-691589 = log H D = log 491-7. 

 Therefore 491-7 — 404 — 87-7 feet. 



Hence B is 87-7 feet above the level of D. 



2nd. When the summit of the station A is below the level of the 

 vertex C of the object C I), fig. 2. 



Let H D = y ; D C = ft' ; the angle H A D = 9 ; and the angle 

 H A C ^= /3 ; and proceeding exactly the same manner as before, 

 we have 



ft' . sin 9 . cos 6 ,.. „ /„i„N /.N 



«' ^ . , , , or = ft . sin 9 . cos 3 . cosec (9 -\- &) (4.) 



" sin (e + P) ' \ ^ J \ I 



A 1 ft . cot /3 . 



And y = r— ; —„ (5.) 



■ cot i3 -f cot 9 ^ ' 



The only difference being in the signs. 



Hence D C -«- A B = height of one object above the level of the 

 other. 



Remark. When the angles are taken to seconds, it is more 

 advantageous to use (2) and (4), as the case may be, to ensure 

 greater accuracy. To solve the foregoing question by the First 

 Case of Plane Trigonometry would recjuire tliereby more than two 

 operations : hence the manifest value of the above formula:'. 



The subject is a valuable exercise to the young student in sur- 

 veying, as giving him jiioper ideas of the utility of trigonometrical 

 formulae for the means of rendering operations more simple iur 

 computation. 



9 — fl = 5" 10'; and A = 300. 



ISOMETRICAL PERSPECTIVE. 



It is to he regretted that isometrical perspective is not in more 

 general use amongst engineers and architects ; but the infrequency 

 of its application to designs of machinery and buildings, arises in 

 a great measure, perhaps, fi-oiu the difficulty usually experienced 

 in properly representing curved lines. In this kind of perspective, 

 we know that a circle inscribed within a square is represented by 

 an ellipse touching the centre of each side of an oblique paral- 

 lelogram, as in the annexed figure. Now, there is no instru- 

 ment which can be set to describe 

 sucli an ellipse, unless we first dis- 

 cover the transverse and conjugate 

 diameters n, b, and c, r/, — a some- 

 wliat troublesome preliminary. It 

 is, therefore, much to be desired 

 that an instrument could be con- 

 trived, whereby the draughtsman 

 would be enabled to produce the 

 ellipse with no more trouble than 

 tlie simple measurement of the 

 radius of the circle to he represented. Perhaps the following 

 suggestion may supply that desideratum : — 



To a block of metal o, let there be two projecting pieces, r and 

 d, each one carrying a pair of radius-bars e,J\ andy, ft ; let those 

 bars exactly agree in length, and let each pair be so united by a 

 pin passing loosely through the projecting piece, that their centre 

 lines form an angle of 30°. Then, if c and g be maile to carry a 

 straight bar ft, and /and ft an arch /, having a tongue or blade j», 

 which bears upon /c, we shall have an instrument by means of 

 which we can describe an isometrical ellipse ; and the mode of 

 using it will be to press upon o, with the thumb of the left hand, 

 and with tlie fingers ot^ tlie same hand turn the arm ?n, and 

 thereby the two pairs of radius-bars — while at the same time the 

 right hand holds a pencil point in the corner o, at the crossing of 

 in and k. The pencil will thus he made to describe half the iso- 

 m etrical representation of a circle, « hose radius is equal to that of 



