18 



MECHANICS. 



tracted ; that square root will be the 

 true weight. Thus, for example, if one 

 counterpoise be 7lbs., and the other 

 9ilbs., the product 7 x O^lbs. is 64, the 

 square root of which is 8. Hence 8lbs. 

 is the true weight. 



To prove this, let a and b be the 

 arms, A and 'B the two counterpoises, 

 and x the true weight. We have 

 x : A : : a : b. 

 B : x ::a:b. 

 Hence we may infer, that 

 x : A : : B : x. 



Independently of finding the true 

 weight, or detecting the fact of a balance 

 being fraudulent, there are several ways 

 in which the design of the vender may 

 be defeated. 



Let the substance to be purchased 

 be bought in two quantities nominally 

 equal, and let these be weighed in different 

 dishes ; the result will be, that the buyer 

 will always get more than the just quan- 

 tity, in proportion as the balance is more 

 fraudulent. Thus, suppose that the 

 arms of the balance were in the propor- 

 tion of 11 to 12, and that two pounds 

 of any substance be weighed in different 

 dishes, the counterpoise in each case 

 being just a pound: in the one case 

 the buyer will receive eleven-twelfths 

 of a pound, and in the other twelve- 

 elevenths ; so that in one portion he 

 will receive one-twelfth less than one 

 pound, and in the other he will receive 

 one -eleventh more than a pound. Now, 

 one-eleventh being more than one- 

 twelfth, he will, on the whole, receive 

 more than the just quantity, by the dif- 

 ference between an eleventh and a 

 twelfth, or by -~^ of a pound. 



In general, let a b be the arms, W 

 the counterpoise, and x and y the two 

 portions, nominally equal, 



t x:W::a:b.-.x= W. J 



y:W::b:a.'.y = W. b - 

 Hence we find, 



Now, the sum of a number and its 

 reciprocal is always greater than 2, and 

 therefore x + y is greater than 2 W. 



But the best way of ascertaining the 

 true weight of a substance with a fraud- 

 ulent balance, or indeed with any com- 

 mon balance, is to place the substance 

 in one dish, and accurately counter- 

 poise it with fine sand in the other. 

 Then take out the substance to be 

 weighed, and replace it by the weights 



with which it is to be compared ; and 

 the equilibrium being produced by them, 

 the true weight will thus be determined, 

 independent of all imperfection of the 

 instrument. 



(44.) Besides the common balance 

 with equal arms, there are various other 

 modifications of the lever used for de- 

 termining the weights of bodies. We 

 have already described one (the Weigh- 

 ing Machine,) for ascertaining very 

 great weights. Those which we shall 

 now describe are suited to ascertain the 

 weights of smaller quantities. 



The Roman balance, or the common 

 steel-yard (fig* 27.) consists of a beam or 

 a bar of iron resting upon knife-edges or 



fiy. 27. 



a pivot, and having one arm much longer 

 than the other. We shall first suppose, 

 that the shorter arm is rendered so heavy 

 as to balance the longer arm when the 

 instrument is unloaded, and that in that 

 state the beam is horizontal. A hook is 

 fixed upon the shorter arm, from which 

 the substance to be weighed is to be 

 suspended, and a determinate and move- 

 able weight P slides on the longer arm. 

 Equilibrium is established by moving 

 the sliding- weight P from the centre G 

 until it acquires such a leverage that it 

 supports the weight W. The arm G B 

 is graduated, so that it indicates the 

 amount of the weight W at the point 

 where the sliding-weight supports it. 



The principle of this machine, and 

 the method of graduating it, are very 

 simple. By the general property of the 

 lever, the condition of equilibrium is, 

 that the weight W multiplied by its dis- 

 tance w from the fulcrum, is equal to 

 the counterpoise P multiplied into its 

 distance p from the fulcrum, or W x w 



