IM 



l.KVKi 



LEVER 



1M 



bstweesj then ; and tbe operation U to be continue. 1 to the end of the 

 line OB which th proMe is required. It U customary to insert the 

 bright* . oe. Ac., in column headed Fart-tiytiii, in a sort of field- 

 book, and the height* AM, n/>, Ac,, in * collateral column headed Bati- 

 tiylHi The difference between the muni of the number* in these two 

 column* will be equal to the height of one extremity of the line above 

 tbe other. 



But it i* very generally the practice, with the view of diminishing 



tb* risk of error truing from the imperfection of the instrument, to 

 execute a tori of double levelling. Thu consists in placing the spirit- 



I succeasively at each of the two station*, at Y and z, and having. 

 by the ecrewi, adjusted the telescope a* before, let I w be the horizontal 

 line at T, and r * that at I ; then, the heighU tr and tx being 

 obtained by means of the staff set up successively at each opposite 

 station, it may be easily proved that half the difference between them 

 will be equal to the height of the ground at one point, as T, above that 

 at the other. This U however strictly correct only when the staves at 

 T and I are considered as parallel to one another ; but the error arising 

 from their being in the direction of the earth's radii is quite insensible 

 in any of the ordinary operations of this nature. 



In using either of these methods, therefore, no correction on account 

 of the earth's curvature is necessary; but when, from any circutn- 

 tanoeB, the spirit-level cannot be placed nearly midway between every 

 two stations, aud particularly when it can be placed only at one station, 

 as T, the difference between the height zr of the visual ray at one 

 station, and T /, the height of the instrument at the other, will not, on 

 account of the earth's curvature, be the correct relative heights of the 

 ground at the two stations. For, let T : be on arc of the earth's surface, 

 supposed to be spherical ; let also Y /, z r, be in the direction of its 

 radii, and let r y be a tangent to the curve at T : then I r being parallel 

 to I g, the difference between z r and Y I, or r y (which may be con- 

 sidered as equal to T t), will be z y, the apparent height of Y above z ; 

 whereas the true height should be z z. Now, from the known magni- 

 tude of the earth, the distance y j, between the tangent Y y and the 

 arc, can easily be computed when Y i or Y z is of any given length. If 

 this length U equal to 100 yards, we shall have y r = 0'02 inches. Con- 

 sequently, in a series of operations, carried on in the manner above 

 described, with station lines not exceeding 100 yards in length, the 

 error in the relative heights at the end of one mile would be little 

 more than one-third of an inch. 



On ascending or descending a steep hill, no other method can be 

 adopted than that of placing the instrument at one extremity of the 

 station-line and the staff at the other ; but as these lines are then 

 necessarily very short, the deviation above mentioned need not be 

 regarded. 



In the determination, on uneven ground, of the length of a base-line 

 for the trigonometrical survey of a country, the relative heights of the 

 ground, as at A, B, c, &c., when found as above, serve for the reduction 

 of the measured hypothenusal lines A B, B c, &c., to the corresponding 

 horizontal line* nn,pq, &c. ; these being comparatively short, are then 

 considered as circular arcs, and each is separately reduced to an arc of 

 the earth's surface at the level of the neighbouring seas by subtracting 



from it the term A-, which i* found from the proportion between the 



arcs and radii in the similar sectors. Here A is the horizontal line 

 or arc, **,; r i* the radius of the earth's curvature at the level of 

 the *ea, and k is the height of the ground at A, B, &c., above that 

 level. 



The profile of the ground is usually expressed on paper, in portions 

 of any convenient length, for the purpose of enabling the engineer to 

 determine the depths of his excavation*, or the height* of the masses 

 of earth to be raised, when it is proposed to execute a canal or road. 

 A right line being drawn to represent one parallel to the horizon, and 

 passing through the highest or the lowest point of the natural ground, 

 the height* or depression* of the remarkable point*, as A, B, &c , with 

 respect to such line, are obtained by additions or subtractions from the 

 nunben in the field-book, and are, by a proper scale, set out from that 

 line on oUiera drawn perpendicularly to it at intervals equal to the 

 horizontal distances between the same point*. The series of points 

 thus obtained, being joined by hand or otherwise, give the figure of 

 the required vertical Motion of the ground, in general, for the sake of 

 dieUnctne**, the Male by which the heights are set out is greater than 

 that of the horizontal distance* between the points. 



n the difference of level only between two place* is required, a 

 rectilinear direction from one to the other i* not necessarily that in 

 which it U moet convenient to perform the operation : a circuitous 

 routs i* preferable when it prceent* fewer impedimenta from wood* or 



level* taken aeraes the lands between the Block and the Caspian seas; 

 and between the Utter and the lake Aral, for the purpose of deter- 

 mining UM relative heighU of those water* : the series which, during 

 tb* expedition of Colonel Cheney, were taken from Iskandcrun on the 

 Mediterranean to Birtbjik on the Euphrates ; and near the Persian 

 'rtween UM latter river and the Tigris. To these may be added 

 the estenive line* levelled in England and on the Continent for the 

 arvenl railway* which have been executed or are in progress; 



and the important work executed under the auspices of the British 

 Association, in order to determine the difference between the levels of 

 the waters in the English and Bristol channels. Thn level* taken by 

 Colonel Lloyd across the Isthmus of Darien, and those taken by the 

 French engineer* across the Isthmus of Suez, may also be referred to, 

 although in both cases the result* obtained hare subsequently been 

 called in question. 



LEVER (Irrart, to lift up), the name of a common mechanical 

 instrument, consisting of a simple bar of wood or metal, by fixing one 

 point of which, called the fulcrum, a pressure at the end more distant 

 from the fulcrum U made to counterbalance a larger pressure at the 

 nearer end ; or if both ends be" equally distant from the fulcrum, equal 

 pressures are made to balance each other. [BALANCE.] 



The lever, considered a* a machine-. Would require no further 

 than a reference to the article PowKn for the correction of a mistake 

 incident to the conception of this and other machines. But as one of 

 the fundamental principles of mechanics receives its most simple form 

 in its application to the common lever, this instrument assumes a 

 degree of theoretical importance which will justify some discussion of 

 the subject : and the principle of the lever, which is often confounded 

 with the lever itself, must be explained. Thus when it is said in 

 popular writings on mechanics that all machines are reducible to the 

 lever and the inclined plane (an assumption of a startling dim. 

 we consider, for instance, the works of a common watch) it is meant 

 that every mode of communicating or relieving pressure is explicable 

 upon the principle of one or other of those machines. 



The first explanation of the lever was given by Archimedes, and 

 that in so simple a manner, that while his method has always been the 

 best for a popular view of the subject, it has never been surpassed, or 

 even equalled, in rigor or purity, considered as a foundation for the 

 science of STATICS. 



It assumes two principles ; firstly, that when a system is in equi- 

 librium, the state of rest will not be disturbed if additional pressures, 

 such as compensate each other, and would by themselves produce no 

 motion, be introduced or removed ; secondly, that when a weight is 

 made to rest by being attached to an immoveable point (say it is sus- 

 pended by a string) the point or pivot of suspension undergoes a 

 pressure equal to the weight of the system, whatever may be the form 

 of that system, or the disposition of its parts. Every science must be 

 founded upon some axiomatic assumption ; and perhaps there is none 



which is better entitled to preference than the fact that a given weight, 

 say a pound, suspended by a string, exerts the same pressure on the 

 string whatever it* shape may be ; namely, a pressure equal to the 

 weight of the body. This being premised, a cylindrical or \m 

 bar of uniform material will necessarily rest if a pivot be passed through 

 its middle section at A : since there is no reason why it should pre- 

 ponderate on either side. Divide the bar into two parts, n c and c D, 

 of which K and I, are the middle points. At K and L suspend . 

 equal to the weights of B c and C ; but at the same tiui. 

 counterpoises' of equal weight acting over fixed pulleys u aud u : so 

 that the new forces being such that each ]>air would W in r<|uililirium, 

 they would not affect the equilibrium already established by means of 

 the equality of the parts of the bar ou . equi- 



librium existing, we are at liberty to remove any forces which equi- 

 librate each other, such as are the upper v and the weight of B ; such 

 also as are the upper w and the weight of o D. For B c, if detached, 

 would exert on the string which goes over the pulley o a pressure 

 neither more nor leas than its own weight (which is v) ; and c D, if 

 detached from the pivot and from B c, would exert ou the string of the 

 pulley u a pressure equal to its own to w. But when these 



pairs are removed, there remain only the lower weights v and w ; the 

 substance and rigidity of the lever being retained to connect them, though 

 its weight is removed or counterpoised. Aud K i., being the R. 

 the halves of the parts, is equal to half of the whole length, or to B A : 

 taek away the common part A K, and there remains B K, equal to A L, 

 or KO equal toAL; also A K is equal to CI-. Also v is t., w in the 

 proportion of B to CD, or of K to c L, or of A I, to A K ; that is, the 

 v and w balance each other wh> inversely as their 



distances from the fulcrum A. 



It only remains to show that no other weight except v, propoi i 

 to W a* above, will counterbalance w. If possible, let anothnr 

 V, produce this effect when applied at K ; and upwards, by means of 

 the pulleys n and o, apply pressures equal to w and v', the old weights 

 v and w remaining as before. Then there are two systems which 



