(iS 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



LMahch, 



3. We shall iKiH 

 vertical section ; tli 

 mitiiiii kvel, and le 

 all given. 



investig.ite an expression for the area of the 

 e inclination of pround, depth, breadth of for- 

 ntrtlis .(, .(', and also the ratio of slope, being 



Fig. 3. 



Through centre O (fiff. .3), draw Pe parallel to A B ; then P O 

 — b+ma. . • . area of trai)ezoid P A B e = (2 4 + m a) « ; 

 area of triangle P O II = i sin e.x' {b + m a) ; 

 and area of triangle D () e = l^sin .r {b + m a). _ 



Consequentlv, the whole area of trapezium or vertical section _ 



area P A B e + area triangle DOe - area triangle P O hl_ 



(26 + ma) a + ^,{b + ma) {x - y)sin8 (3) 



The first column of the table gives the angle of inclination of 

 the ground, and the adjoining column the nat sines to three places 

 of decimals, to facilitate working out the area, as in equation (3). 

 We shall now commence with the following Rules. 



I To find thf. two lengths O D and O H :— Rule. Add the half- 

 breadth of formation level to the product of the slope and given 

 depth • then multiply this sum bv the corresponding tabular num- 

 ber, then will each product be equal to each length required. 



II. To find the area of section H A B D :— Rule. 1st. Add the 

 formation level to the product of the ratio and depth, and midtiiily 

 this sum by the depth. 2ndly. Add half the formation level to the 

 product of ratio and depth ; multiply this sum by the difference of 

 the two lengths, and again by nat sine of angle. Add these two 

 products, and their sum will be the area. 



E.ramp!e 1.— Given the angle of inclination of ground 18°; slope, 



1 to 1 ; depth, 45 feet ; and breadth of formation level, 30 feet. To 

 find distances of centre stake, area of section, and cubic content, 

 when 100 feet in length. 



Here ft -I- a m = 15 -H 45 = 60 ; m = 1 ; 6 = 18°; its nat sin = -309 



. • . 1-557 X GO = 93-429 = O D. -799 X 60 = 47-940 = O H. 



By formula (3) we have (30 + 45) 45 -t- i (15 -(- 45) (45-48y309 



= 75 X 45 + 30 X 45-48 X '309 = 4099-5 area required. 



Cubic content = 409950-0. 



Example 2.— Given angle of inclination of ground, 20° ; slope, 



1| to 1 ; depth, 50 feet ; and breadth of formation level, 30 feet. 



To determine distances and also area. 



Here a = 50 ; 6 = 15; m- l^ = * ; fl = 20° ; its nat sin = -342 



. - . 6 -f a m = 15 + 75 = 90. 

 Xow, 2-344 X 90 = 210-96 = O D. -781 X 90 = 70-29 = O H. 



By formula (3) for area we have 

 (30 -I- 75) a -f- 4 (15 + 75) (140-67) -342 = -7174 area required. 

 Example 3 — Given the inclination of ground, 18° ; slope to be 



2 to 1 ; depth from field-book, 20 feet ; breadth of formation level, 

 30 feet. To find area and distances. 



Here 6 = IS ; a = 20 ; fl = 18 ; m = 2. . • . 6 -j- om = 55. 

 55 . X 3-000 = 1 65- = O D. 55 . X -641 — 35-25 = O H. 

 By formula (3) we have 

 (30 4- 40) 20 -f ^ (15 -f- 40) (129-74) -309 = 2505 area required. 



Remark. — If the ground should ascend and descend, as in the 

 adjoining diagram (fig. 4), then Table No. II. is to be used to find 

 the distances. Table No. I will in like manner be required for 

 ground descending from centre, as in fig. 5. 



Fig. 4. 



•Fig- -I- 



The Tables will likewise do for embankments — No. I. for the 

 TRscent from centre stake, and No. II. for the descent. 



We shall now discuss the equations (1) and (2). Put them 



respectively under the following forms. T, T', being tabular num- 

 bers, A = 6 4- m a. 



X = T . A ; and X' = T' . A. 



Divide by T, T', respectively ; then y„ = 'f^, A being eliminated. 



Therefore the two distances, ,r and .r', are to each other as their 

 respective tabular numbers; consequently, the distances can be 

 proved by a second operation. The Tables might have beeii car- 

 ried up to 45°, but then they would r.^|uir<; a greater uuinber of 

 places of decimals to insure greater accuracy. 



In taking tlie angle of inclination, the climmjter or common 

 theodolite might be used ; but if the spirit-level should be used, 

 then we have only to measure from O downwards nn^distanoe. Or, 

 (fig. 6), and then take the height with instrument ; and then will 



,. . I,.. ,. •/-., r\ 1 height 



the sine oi angle ot inclination U as = rUa = t^ — ~ — . 



distance 



Erratum, — The diagrams, figs. 2 and 6, in the aboi;e article, are 

 transposed, for which oversight the printers are accountable ; but 

 beyond such transposition the error does not extend. 



REVIEWS, 



The Port and Docks of Birkenhead ; irith Maps, Plans, Sections, 

 and Tidal Diagrams, and an account of the Acts of Parliament re- 

 lating to the Mersey and Dock Estate of Liverpool. By Thomas 

 Webster, M.A.. F.R.S. London : Chapman and Hall, 1848. 



Birkenhead lias been a wonder, and has had its nine dajs, and very 

 many are quite ready to believe that we have had enough of it. 

 The announcement of sucli a town springing up in England was 

 calculated to creiite as much astonishment as that of Aladdin's 

 palace fresh coined by liis wonderful lamp. It is not so easy to 

 create great wonders in an old and settled country like this : cities 

 of whitewash and timber-framing, metropolitan centres of slab- 

 houses and log-huts, we leave to the far west of the States, or the 

 sandy regions of Australia ; and we should be no more surprised 

 by the flourishing account of a Babylonian capital newly hatched 

 in California, than by the siglit of the three last joints of the sea- 

 serpent's tail, or the repudiation of a fresh batch of Pennsyhanian 

 bonds. Towns grow in the \irgin soils of the new world ; they 

 are a natural production — or at any rate they can be planted as 

 easily as cotton, or what the Americans dignify with the name of 

 corn. We can reconcile ourselves to such creations as Fleetwood, 

 or Kingston-upon- Rail way, Wolverton, or Swindon, — the resuscita- 



