B A L 



B A L 



the Crft divifion fhall equiponderate with one ounce in 

 the fcale ; and llie body to be weighed be put in the 

 fcale, and the above mtn1.io;.td weight be moved along 

 the longed brachium, till the eqnillbiium be found ; the 

 number of divifions between the body and the centre fliews 

 the number of oiificcs tliat the body weighs, and the 

 fubdivifions the parts of an ounce. On the fame princi- 

 ple alfo is founded the Jecei'.Jul balance, which cheats by 

 the inequality of the brachia : for inftance, take two fcales 

 of unequal weights, in the proportion cf 9 to 10, and 

 one of them at the tenth divifion of the balance above 

 dcfcribed, and another at the ninth d-vifion, fo that there 

 may be an equilibrium ; if then you take any weights, 

 .which are to one anothtr as 9 to 10, and put the firft 

 in the firll fcale, and the fecond in the other fcale, they 

 will equiponderate. 



But it is eafy to difcover the deceit of a falfe balance by 

 changing the weights thiit are in eqiiihbrio to the contrary 

 fcales ; and thus the owner of the balance mull either con- 

 fefs the fraud, or add to the commodity fold by means of 

 fuch a balance, not only the quantity by which it was defiei- 

 ent, but alfo as much as he intended to gain by the fraud, 

 and a fraAion of that added weight proportional to tlie ine- 

 quality of the arms of the balance. In tliis ca(e, the buyer, 

 inftead of C)\\, offered to him fo;- lotb his due, will have by 

 changing the fcales, iii pounds. For 9 : 10 :: 10 : 1 1^- 



Several weights, hanging at feveral dillances on one fide, 

 .may equiponderate with a lingle weight on the other fide : 

 to do this it is required, that the produdt of that weight, by 

 its diftance from the centre, be equal to the fum of the pro- 

 duels of all tiie other weights, each being multiplied by its 

 diftance from the centre. 



To demonftrate which, hang three weights of an ounce 

 each, at the fecond, third, and fifth divifions from the cen- 

 tre, and they will equiponderate with the weight of one fin- 

 gle ounce applied to the tenth divifion of the other brachium ; 

 and the weight of one ounce at the fixth divifion, and 

 another of three ounces at the fourth divifion will equi- 

 ponderate with a weight of two ounces on the other lide 

 at the ninth divifion. 



Several weights unequal in number on either fide, may 



equiponderate : in this cafe if each of them be multiplied 



-by its diftance from the centre, the funis of the products 



cn either fide will be equal ; and if thofe fums be equal, 



there will be an equ'dibrium. 



To prove which, hang on a weight of two ounces at the 

 fifth divifion, and two others, each of one ounce, at the fe- 

 cond and fevcnth ; and on the other fide hang two weights, 

 each alfo of one ounce, -at the ninth and tenth divifions; 

 and thcfe tvio will equiponderate with thofe three. A ba- 

 lance of this kind, the arms of which are equally divided, 

 has been fometimes called an anthmctlcal balance ; becaufe 

 the arithmetical operations of addition, fubtraction, multi- 

 plication, and the rule of three, may be cafily performed by it. 



E. g. To add i\\e. numbers 2, 3, and 7; apply an ounce 

 weight at the fecond divifion, and another on the fame arm 

 at the third, and another at ttiefeventh, then take an ounce 

 weight, and move it along the other arm, till the beam is in 

 cquilibrio, which will be at the twelfth divifion; fo that 

 2+3 + 7 = 12. 



To fubtraU 5 from I 2 ; hang an ounce w eight at one end 

 of the arm at i z inches, and another at the other end at 5 ; 

 then move a third ounce weight along the arm till the equi- 

 librium is reftored, and it will be found at the feventh divi- 

 ifion, which gives 12 — 5 = 7. 



To multiply 4 by 3 ; fufpend a four ounce weight at the 

 ■sJiird divifion on one arm, and move an ounce weight on the 



other, till the beam be in equilibrio, and it will mark out 

 I2=4V3. 



Tp div'idc 12 by 4; fufpend an ounce at the twelfth divi« 

 fion, and move a four ounce weight on the other arm, till 

 there is an equilibrium, a"d it will be found at the quotient 

 3=;\j^. s'Gravefaiide. Phyfices Elem. Math. vol. i. p. 50. 



To the juftnefs of a balance it is required, that the points 

 of fufpeiifion be cxadlly in the fame line as the centre of 

 the balance; that they be precifely equidltlant from that 

 ce:.tre on cither fide ; that the brachia be as long as conve- 

 niently they may, in relation to their thick nefs, and the 

 weight which they are intended to fupport; that there be 

 as little fricli.;ii as pofQble in the motion of the beam and 

 fcales; and lallly, that the centre of gravity of the beam 

 be placed a little below the centre of motion. 



We (lull here add fome further obfervp.tions, which may 

 ferve to iUuftrate thefe properties of a good balance, and 

 which deftrve attention in the conftruClion of this inftrument 

 for purpofes that require peculiar accuracy. The balance 

 is properly a lever, wliofe axis of motion is formed with an 

 edge like that of a knife, and the two dilhes or fcales at its 

 extremities are hung upon edges of the fame kind, which 

 are firft made lliavp, and then rounded with a fine hone, or 

 a piece of butf leather. On the regular form of this 

 rounded part the excellence of the inftrument very much 

 depends. When the lever, or beam of the balance, is 

 cotifidered as a mere line, the two outer edges are called 

 p.ointsof fufpenfio!!, and theinnerthe fulcrum. The points of 

 fulpenfion are fuppofed to be at equal diftances from the 

 fulcrum, and to be prelfed with equal weights when loaded. 

 I. If the fulcrum -be placed in the centre of gravity of the 

 beam, and the three edges be all in the fame right line, the 

 beam of the balance will have no tendency to one potitiou 

 more than another, but will reft in any pofition in which it 

 may be placed, whether the fcales be on or off, empty or 

 loaded. 2. If the centre of grajity of the beam, when le- 

 vel, be immediately above the fulcrum, it will overfet by the 

 fn.allert action; that is, the end which is loweft will defccnd; 

 and it will do this with the greater velocity, in proportion 

 as the center of gravity is higher, and the points of iufpen- 

 fion are lefs loaded. 3. But if the center of gravity of the 

 beam be immediately below the fulcrum, the beam will not 

 reft in any pofition but when level ; and, if difturbed from 

 that pofition, and then left at liberty, it will vibrate, and at 

 laft come to reft in an horizontal pofition. Its vibrations 

 Will be quicker, and its horizontal tendency ftrongei", the 

 lower the centre of gravity, and the lefs the weight upon the 

 points of fufpenfion. 4. If the fulcrum be below the line 

 joining the points of fufpenfion, and thefe be loaded, the 

 beam will overfet, unlefs prevented by the weight of the 

 beam tending to produce an horizontal pofition, as in the 

 third cafe. In this laft cafe fmall weights will equilibrate, 

 as in the laft cafe ; a certain exaft weight will reft in any po- 

 fition of tiie beam, as in the firft cafe ; and all greater weights 

 will caufe the beam to overfet, as in the fecond cafe. Mo- 

 ney fcales are often made this way, and will overfet with any 

 confiderablc load. 5. If the fulcrum be above the line 

 joining the points cf fufpenfion, the beam will come to the 

 horizontal pofition, unlefs prevented by its own weight, as 

 in the fecond cafe. If the centre of gravity be nearly in 

 the fulcrum, all the vibrations of the loaded beam will be 

 made in times nearly equal, unlefs the weights be very fmall, 

 when they will be flower. The vibrations of balances are 

 quicker, and tiie horizontal tendency ftronger, the higher 

 the fulcrum. When the fulcrum, or centre of motion C, 

 (fee _/^'. 10.) is in the right line joining the centres of fuf- 

 penfion, it is evident that the equilibrium of equal weights, 

 e. g. Pand W, will obtain in every pofition ; for the perpen- 

 diculars 



