434 Prof. Waxuace on the Invention of the Pantograph, 
« Annex two ciphers to the difference of the numbers which ex- 
press the proportion, and divide the result by their sum ; the 
quotient is the number on the scale which must be set opposite 
to the index on the socket.” Thus, to give the proportion of 3 
to 2, the difference of which, with two ciphers annexed, is 100, 
divide 100 by 5; the quotient 20 on the scale must be set oppo- 
site to the index. Ifthe proportion be that of 5 to 3, we have 
for the dividend 200, and for the divisor 8; this gives 25 for 
the division to be set at the index; and so on. In general, the 
quotient is a fraction, but the nearest whole number is accurate 
enough in practice. Thus, if a design is to be reduced in the 
proportion of 8 to 5, the sum being 13, and the difference, with 
two ciphers annexed, 300 ; the number on the scale to be set to 
the index is %2,° = 23, nearly. In some instruments a vernier 
scale has been engraved on the socket, by which each half of the 
beam was actually divided into 1000 equal parts ; and then, be- 
tween the proportion of equality and that of 6 to 1, there might 
be interposed *3°° = 714 different ratios, namely, those of 1001 
to 999, of 1002 to 998, &e. 
Tue wHeets ff. These turn on steel axes which pass 
through vertical bored tubes fixed to the ends of the beam. The 
diameter of each is about four inches, and they ought to be exactly 
equal, because on this the accuracy of the strument mainly de- 
pends. 
There is a portion of each wheel, about a third of an inch in 
breadth from its circumference, which is thicker than the part 
within it. This is shewn in Fig. 5, which is a section of a wheel 
through its centre ¢ and a part of the beam, of half the actual 
size. The upper surface of this bounding ring rises higher than 
the surface it incloses; and the lower surface also rises higher 
than the lower central surface. Thus, additional space is given 
for the groove in the circumference in which the band lies, and 
the band thus raised is kept quite clear of the arms which pass 
freely below it. 
