402 On the Graduation 
must be properly done. I have not seen a sufficient number of 
the larger instruments to be able to speak positively on this point, 
but [ believe that in general the above requisites are not com- 
plied with. 
In a two-foot instrument of excellent workmanship, which now 
lies before me, there is a scale of 100 parts on oc bar; that is, 
the space between the extreme marks 4 and +1, is divided into 
100 equal parts, zero’ being against the upper end. The want 
of principle manifested by this arrangement is obvious; for it 
may be demonstrated that, whether the pencil be attached to the 
long or to the short bar, that scale only which is adapted to it 
ought to contain equal divisions; and ‘the other, or scale for 
the fulerum, must be unequally graduated. The scales, there- 
fore, on the instrument alluded to, are worse than useless, for 
they may mislead the practitioner, in case he should forget the 
fundamental rule, that in all cases the three operative points 
must be ina right line. , 
When the ratio of reduction is greater than 4, the pencil must 
be attached to the long bar, and the fulcrum to the short one. 
Again, when the ratio is not greater than 2, it will be proper 
that the pencil and fulcrum should change places. Each of 
these cases requires a distinct scale for each ‘bsr. 
The scales applicable to the first case, [ shall call A, and their 
divisions will be numbered in each bar from 100 to 50. Those 
applicable to the second case, | shall call B, and they will be 
numbered from 50 down to 8, or lower, if the mechanism will 
admit of it, though a less ratio than 100 to 8 will rarely be 
wanted in practice. 
Let the ratio of the original to the copy be represented by 
1:7, and let 2 d= the working length of the instrument, that 
is, the distance from the ver tical pivot to the centre of the tube 
that carries the tracer, or to the Jowest mark on the left-hand 
dar: consequently d= the distance between the two pivots on 
the last-mentioned bar, and equal to the distance from the pivot 
at the upper end of the short graduated bar to the lowest mark 
on it. Then the distances of the several divisions, to be laid 
off feom the pivot connecting the graduated bars, will be as fol- 
lows: 
dr on the long bar. 
For the scale A < 2dr 
7 on the short bar. 
dr 1 ‘ 
pitas ciate net {= on the long bar. 
2dr on the short bar. 
According to the above formule the subjoined table’ is con- 
structed, assuming 2d = 1; consequently, to apply the table 
Lo" 
