1846.] 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



27: 



minute errors from the necessary omission of small decimals. — For ex- 

 ample : On a gradient of 1 in SOO, the value of G for 50 feet would be 

 0'1666G, &c.; now, if we used 0'17 as the nearest approximation and 

 repeated it, we should obtain a total difference of 17 feet in one hundred 

 lengths of 50 feet ; whereas the correct difference for 5000 feet is 16 feet 

 8 inches. But if we take G, for the first two lengths as 0-17, and for the 

 third 016, and pursue the same alternation for the entire length, we shall 

 have a correct result. This will, it is hoped, be sufficient to show one 

 subject deserving of care. All the points laid out should be marked by a 

 nick in the ground ; but if such a mark is not very visible, small wooden 

 pegs must be employed. 



The cross sections annexed exhibit what has been done so far, and what 

 there still remains to do. 



-Embankment. 



There has been determined, the height of embankment or depth of cut- 

 tingcc'; the width, measured horizontally, of roadway cd, cd', and for 

 slopes de, d'e' ; and the heights ch, fe, /',' have been taken. And there 

 remains to calculate, the horizontal addition to the widths for slopes e x on 

 one side, and horizontal deductions on the other side, e'x'. The properties 

 of the similar triangles cge, cxx, and of xxg, agd, afford a variety 

 of expressions for the distance at g- ; and in making a selection from them, 

 I have chosen that which requires the least quantity of field work, without 

 increasing the labour in the office. 



The notation employed will be 



Kg. 3.- 



Form of Level Book to be 



C, to represent the height of embankment, or depth of cutting. 



r = the ratio of the slopes. 



B=half the width of roadway, ac, or c d. 



k = the height read off the staff at the centre, c h. 



h'= the height read off at the edges of the slopes as first set out 



that is/ < and/' e'. 



Take H = A — A'. The expression then for the horizontal distance 



(r-C' + B)H 



gxia its general form is r.< i B „" 

 ^ -r — — 11 



r 



In applying this formula we must pay attention to the essential signs of 

 H and — , as well as of C. 



As already explained, the essential sign of H is positive when the ground 

 m cross section is rising, and is negative when it is falling; and the sign of 



— is to be taken as negative in embankments, for the slopes fall from 



the centre towards either side, — and as positine in ccttings, because the 

 slopes rw« in the same manner ; the sign by which the result is affected 

 will show whether it is to be added to, or deducted from the width already 

 set out for slopes. 



The calculation of this formula is best performed in the office ; and the re- 

 sult in each case should be entered in the proper columns of the level-book, 

 to be set out on the ground on the next day devoted to field work, when the 

 marks indicating the spaces for slopes should be removed, and replaced 

 after the corrections. 



It is recommended to use the ordinary slide rule in making the calcula- 

 tions—a modification, in the shape of two concentric circles of card board, 

 divided on the edge into a logarithmic scale, of which the circumference 

 of the smaller circle is the logarithm of 10, I have employed with much 

 satisfaction. The outer circle is laid down on a rectangular piece of card- 

 board, and divided from right to left ; the inner circle is a separate card, 

 cut exactly round its circumference, is divided from left to right, and is 

 moveable upon the rectangular card on its centre ; and all the calculations, 

 involving multiplication and division, are performed by inspection. The 

 smaller circle may convenintly be 3J or 3J inches in diameter. 



I need not encumber the Journal with a demonstration of the simple 

 formula; I have given; the investigation may, however, prove an amuse- 

 ment to the younger members of the Profession, and to them it will be a 

 useful employment for half-an-hour. The only postulate is that the three 

 points, c, e, x, should be in one straight line lying in the plane of the cross 

 sections. 



When the corrections to the widths for slopes are made which are due to 

 the inclination of the ground surface, we complete the work by adding the 

 necessary width for hedge and ditch, which should be measured off hori- 

 zontally ; this offers no difficulty. 



When all the widths are measured off, the several lines are marked on 

 the ground by nicking it with a spade or grafting tool, or by running a. 

 light furrow with a plough over the several marks. 



A convenient form of level book for entering the particulars, which 

 ought to remain on record, belonging to the setting out of the widths, has 

 been alluded to in a previous part of this paper. I annex a form of this 

 kind, and have inserted in it, the registry of a short length of r&ilway, suf- 

 ficient to offer a praxis, to those disposed to pursue the subject. 

 used for Setting out Widths. 



1, Lancaster Place, Aug-ust I, 1846. 



J. H. 



36 



