

The principle of momenta ia true for each kind, and therefore 

 this reaaon I have above avoided, 



statement of tho <ic.u>ral principle, the terms "long arm" 

 and " matead " power arm " and " reist- 



anoo arm," indicating thereby the arms that work with tho 

 power or idth the resistance. 



The example of a combination of levers which in most likely 

 to interest you, is tho common u'dyhiwj machine, used for 

 weighing loaded markpt carte, or luggage at railway station*. 

 In I .l:in of 



I:IM piece of moohanism, whore 

 at A, n, c, D, tho four corners 

 of tiio bottom of a shallow box, 

 are the fulcrum* of four levers 

 of tho eooond ordor, which meet, 

 two and two, on cither side at 

 F, and aro joined across by a 

 stout steel pin, by which they 

 arc also connected with tho lever 

 of the second order, E o, which 

 has its fulcrum at E. Tho end, 

 o, of this lever is connected by 

 a rod which ascends perpondi- 



Fig. 52. 



cnlarly from the ground, and is attached above to the short 

 arm of another lever one of the first order, generally a steel- 

 yard, to be afterwards described to the longer arm of which tho 

 weighing counterpoise is attached. We thus have a triple com- 

 bination of levers, the firsb four at the bottom, by being united 

 at F, being virtually one lever. On these four at a, 6, c, d, are 

 four points of hardened steel, presented upwards, on which rests 

 the square wooden platform, on which the cart or luggage to be 

 weighed is placed. Tho weight pressing at a, b, c, d, tends to 

 depress the common end, F, of the four levers, and with it also 

 the end, o, of the lever E F G. The latter tries to pull down the 

 rod, and with it the short arm of the steelyard above, which 

 pull is resisted by the counterpoise on tho longer arm of tho 

 steelyard, producing equilibrium, and making known the weight 

 of the cart or luggage. 



For example, taking the four platform-levers as one, suppose 

 the resistance arms in tho combination arc each one-fifth of the 

 power arms, then evidently, as proved above, the resistance is 

 5 multiplied three times into the power that is to say, 1 pound 

 above on the steelyard balances 125 pounds, or 1 cwt. and 13 

 pounds on the platform. If tho proportion wero one-eighth, it 

 would balance 4.1 cwt. 8 pounds, which strikingly illustrates tho 

 mechanical advantage gained in these machines. We will now 

 consider the common balance, and, in the next lesson, examine 

 the principles of other weighing instruments, bent levers, and 

 tho wheel and axle, and their combinations. 



THE COMMON BALANCE. 



Of weighing instruments, the scale, or common balance, 

 claims the first attention. It is a lever of tho first order, in 

 which the counterpoise, or power, is equal to the resistance, or 

 substance weighed. There is first the beam, A B, at tho ends of 

 which (Fig. 53) are tho hooks, from which hang the chains or 

 oords which support the pans or scales below. Since tho 

 weights in the scales are required to be equal, the fulcrum, F, 

 should be in the middle of the beam, equally distant from the 

 points of suspension of the chains, elso the balance is fraudu- 

 lent, for the purchaser who has his tea or sugar served to him 

 from the end of the longer arm is getting less than his money' e 

 worth. I shall direct your attention to the case in which the 

 line joining the points, A B. of suspension passes through the 

 supporting point of tho fulcrum, as it is tho simplest ; and 

 balances of this kind, as you will see, have a peculiar advantage 

 as to their sensibility. 



Now, it is evident, since A B is bisected at F, and tho scales, 

 chains, and weights on either side aro equal forces, that what- 

 ever be the position in which I place the beam, the resultant of 

 these forces must pass through F, and, being there resisted, 

 leave the whole apparatus at rest. Moreover, if tho centre of 

 gravity of tho beam is at F, so far as its weight is concerned, 

 there will be equilibrium in every position. But such a pair 

 of scales would be utterly useless, since, for equal weights, tho 

 arms should rest only in an horizontal position. 



How, then, is this latter object accomplished ? By having 

 the centre of gravity of tho beam below the fulcrum, when tho 



armH are horizontal. The desired petition U then one of ttaJblo 

 equilibrium (see Lesson VII.), to which the beam will revert 

 -placed from it, and in which the line F o is perpendi- 

 cular to the line A B, joining the potato of suspension of the 

 scales. For a good pair of scale*, therefore, there must be 

 liability as well M accuracy. 



But a balance should also be icntitice should indicate ft 

 light difference of weights in the scale*. How U this secured ? 

 Suppose tho scales equally loaded, and that a slight additional 

 weight (call it P), ia thrown into the scale a in Fig. 53, 

 causing it to decline through some angle agreed upon M suffi- 

 cient to indicate a difference of weights to the eye. As A de- 

 scends, the centre of gravity, o, of the beam ascends at the other 

 side, until it weight (call it w), acting at o, balances i 

 have thus a new lover, A D, tho fulcrum of which also is F, and 

 at whoso ends the forces p and w act. And since in that case, 

 i as proved in the last lesson, p multiplied by A F must be equal to 

 ; w multiplied by F D, the length A B, and the weight w, of the 

 beam being the same in any number of balances in a manufac- 

 tory, that one which moves through the angle agreed on, with 

 tho smaller additional weight p, must sJao hare F D smaller; or, 

 which comes to tha same thing, since the angles of the triangle 

 F a D are given, that at F being a right angle, it must hare F o 

 smaller. Everything else, therefore, being the same, that balance 

 has the greater sensibility, the centre of gravity of whose beam 

 is as little as possible below the fulcrum. Summing up, then, we 

 have for the requisites of a good balance the following : 



1. For Accuracy. That the arms be equal. 



2. For Stability and Horitontality. That the centre of 

 gravity o^ the unloaded beam be below the fulcrum, on a line 

 through its supporting point, perpendicular to that which joins 

 the points of suspension of tho scales. 



3. For Sensibility. That tho centre of gravity of the beam be 

 as little as possible below the fulcrum. 



You will observe that the second and third conditions oppose 

 each other. The lower the centre of gravity is below the 

 fulcrum, the greater is its stability, but the less its tennbility. 

 Both qualities aro essential, and are therefore secured only by a 

 compromise; the centre for sensibility may approach the ful- 

 crum, but not too close ; for stability it keeps off, but not too 

 far. 



Further, observe the consequence of making the line joining 

 the points, A B, of suspension pass through tho fulcrum. How- 

 ever the pans are loaded, 

 it is only the difference (r) 

 of the weights in them that 

 affects the sensibility. Tho 

 resultant of the lesser one 

 in B, and of as much of that 

 in A as is equal to it, passes 

 through and is resisted by 

 F, and affects neither stabi- 

 lity nor sensibility. If A B 

 were not to pass through 

 F, then these weights would 

 have influence as regards 

 these qualities, but that 

 kind of balance we are not 

 hero considering. 



A most important qnes- a pj g 53. 



tion is, how to detect fraud 



in a pair of common scales. The arms in that case not being 

 equal, all tho purchaser has to do, if he doubts the honesty of 

 hia tradesman, is, after tho first weighing, to make the shop 

 weight and the substance weighed change pans. If the two 

 balance each other equally as before, the scales are honest 

 the arms are equal ; but if not, fraud is proved. 



But how, in that case, may the purchaser still get his true 

 pound of tea, or sugar, or other commodity r The shop weight 

 being supposed true, tho imperial stamped weight, let the 

 deficient tea be weighed as before from the longer dishonest 

 arm. Leaving it then in the scale, let him require the shopman 

 to remove the weight from tho other scale, and fill it with tea 

 until that in the first one is balanced. He now has a true 

 pound of tea balancing the deficient pound, as the imperial 

 weight first did. Lot him carry off this pound, and he has his 

 money's worth. 



But there ia another way by which the purchaser may not 



