320 



THE POPULAR EDUCATOR 



counterpoise is attached. We thus have a triple combination 

 of levers, the first four at the bottom, by being united at F, 

 beir.g virtually one lever. On these four, at a, b, 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. The weight 

 pressing at o, b, c, d, tends to 

 depress the common end, F, of 

 the four levers, and with it also 

 the end, G, 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 the longer arm of the steel- 



Fig. 52. 



yard, 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 the combination are 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 the proportion were one-eighth, it 

 would balance 4 cwt. 8 pounds, which strikingly illustrates the 

 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 

 the 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 the 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 the hooks, from which hang the chains or 

 cords which support the pans or scales below. Since the 

 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, else the balance is fraudu- 

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

 from the end of the longer ana is getting less than his money 'P 

 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 the fulcrum, as it is the 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 the scales, 

 chains, and weights on either side are 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 ret. Moreover, if the centre of 

 gravity of the 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, the 

 arms should rest only in an horizontal position. 



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

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

 arms are horizontal. The desired position is then one of stable 

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

 when displaced from it, and in which the line F G is perpendi- 

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

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

 stability as well as accuracy. 



But a balance should also be sensitive should indicate a 

 slight difference of weights in the scales. How is this secured ? 

 Suppose the scales equally loaded, and that a slight additional 

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

 causing it to decline through some angle agreed upon as 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 its weight (call it w), acting at G, balances p. We 

 have thus a new lever, A r>, the fulcrum of which also is F, and 

 at whose ends the forces P and w act. And since in that case, 

 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 

 the smaller additional weight p, must also have F D smaller; or, 

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

 F G D are given, that at F being a right angle, it must have F G 

 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 Horizontally. That the centre of 

 gravity of the unloaded beam be below the fulcrum, on a line 

 through its supporting point, perpendicular to that which joins 

 the points of suspension of the scales. 



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

 as little as possible below tho 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 stabilHy, but the less its sensibility. 

 Both qualities are 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 the fulcrum. How- 

 ever the pans are loaded, 

 it is only the difference (p) 

 of the weights in them that 

 affects the sensibility. The 

 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 

 here considering. 



A most important ques- 

 tion is, how to detect fraud 

 in a pair of common scales. 



Kg. 53. 



The arms in that case not being 

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

 his tradesman, is, after the 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 ? The shop weight 

 being supposed true, the 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 the 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. Let him carry off this pound, and he has his 

 money's worth. 



But there ia another way by which the purchaser may not 

 only get his due quantity, but turn the tables on the vendor, 

 and by the very fraudulent balance itself get more than his 

 money's worth. Suppose he is buying two pounds ; then let 

 him have one pound weighed in one scale and the other pound 

 La the other scale ; it so happens that invariably the two to- 

 gether are more than two pounds. The reason you will under- 

 stand by an example. Suppose one arm is 14 inches long, 

 and the other 15 inches. Then, weighed at the latter arm, the 

 purchaser gets only Hths, which is less, but at the former ifths, 

 which is more than one pound. But by the latter he gains a 

 i^th of a pound more than he is entitled to, while at the former 

 he ..oses only ^th. So on the whole, since a tVth is greater 

 than a -^th, he is a gainer ; he has caught the vendor in his 

 own trap. Or, you may add up the two fractions ifths and 

 Hths, and the sum you will find to be greater than 2 by the 

 fraction ^5. And what is true of these numbers is true of 

 all others, which represent the proportion of the arms what 

 you lose at the long arm is more than recompensed by what 

 you gain at the short one. # 



